| Qubit IDs | \r\n", + "0, 1, 2, 3 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|0000\\right\\rangle$ | \r\n", + "$0.3536 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010\\right\\rangle$ | \r\n", + "$0.3536 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101\\right\\rangle$ | \r\n", + "$0.3536 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110\\right\\rangle$ | \r\n", + "$0.3536 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000\\right\\rangle$ | \r\n", + "$0.3536 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011\\right\\rangle$ | \r\n", + "$0.3536 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100\\right\\rangle$ | \r\n", + "$0.3536 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110\\right\\rangle$ | \r\n", + "$0.3536 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "49956 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 2916 physical qubits to implement the algorithm logic, and 47040 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "50us 400ns | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (3us 600ns) multiplied by the 14 logical cycles to run the algorithm. If however the duration of a single T factory (here: 36us 400ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "18 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 5\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 18$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "14 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 2 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 2 CCZ and 2 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "14 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 14. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "16 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 2 CCZ and 2 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 16 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 12 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{16\\;\\text{T states} \\cdot 36us 400ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 50us 400ns\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "2 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 16 T states, the 12 copies of the T factory are repeatedly invoked 2 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "2916 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 2916 are the product of the 18 logical qubits after layout and the 162 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "47040 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 3920 physical qubits and we run 12 in parallel, therefore we need $47040 = 3920 \\cdot 12$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "1.98e-6 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 18 logical qubits and the total cycle count 14. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "3.13e-5 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 16. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "9 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.000001984126984126984)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "162 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "3us 600ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-7 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{9 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 3us 600ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 162. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "3920 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "36us 400ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.13e-5. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "7 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "3920 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "36us 400ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.13e-5 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 3.13e-5. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "5 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 18 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "2 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "2 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "2 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "| Qubit IDs | \r\n", + "0, 1, 2, 3, 4, 5 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|000000\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110\\right\\rangle$ | \r\n", + "$0.1768 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "84536 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 6776 physical qubits to implement the algorithm logic, and 77760 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "277us 200ns | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (4us 400ns) multiplied by the 63 logical cycles to run the algorithm. If however the duration of a single T factory (here: 46us 800ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "28 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 9\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 28$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "63 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 12 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 5 CCZ and 12 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "63 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 63. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "68 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 5 CCZ and 12 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 68 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 12 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{68\\;\\text{T states} \\cdot 46us 800ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 277us 200ns\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "6 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 68 T states, the 12 copies of the T factory are repeatedly invoked 6 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "6776 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 6776 are the product of the 28 logical qubits after layout and the 242 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "77760 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 6480 physical qubits and we run 12 in parallel, therefore we need $77760 = 6480 \\cdot 12$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "2.83e-7 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 28 logical qubits and the total cycle count 63. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "7.35e-6 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 68. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "11 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.0000002834467120181406)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "242 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "4us 400ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-8 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{11 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 4us 400ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 242. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "6480 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.17e-6. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "9 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "6480 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.17e-6 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 7.35e-6. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "9 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 28 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "5 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "| Qubit IDs | \r\n", + "0, 1, 2, 3, 4, 5, 6 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|0000000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001110\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010110\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100110\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111000\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000110\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011000\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100010\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100100\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101000\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110000\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111001\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111011\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111101\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111111\\right\\rangle$ | \r\n", + "$0.0000 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "90290 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 6050 physical qubits to implement the algorithm logic, and 84240 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "616us | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (4us 400ns) multiplied by the 140 logical cycles to run the algorithm. If however the duration of a single T factory (here: 46us 800ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "25 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 8\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 25$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "140 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 20 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 20 CCZ and 20 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "140 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 140. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "160 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 20 CCZ and 20 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "13 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 160 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 13 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{160\\;\\text{T states} \\cdot 46us 800ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 616us\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "13 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 160 T states, the 13 copies of the T factory are repeatedly invoked 13 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "6050 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 6050 are the product of the 25 logical qubits after layout and the 242 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "84240 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 6480 physical qubits and we run 13 in parallel, therefore we need $84240 = 6480 \\cdot 13$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "1.43e-7 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 25 logical qubits and the total cycle count 140. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "3.13e-6 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 160. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "11 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.00000014285714285714285)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "242 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "4us 400ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-8 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{11 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 4us 400ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 242. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "6480 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.17e-6. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "9 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "6480 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.17e-6 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 3.13e-6. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "8 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 25 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "20 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "20 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "20 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "| Qubit IDs | \r\n", + "0, 1, 2, 3, 4, 5, 6, 7 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|0000000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00011110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00110100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00110110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00111000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00111010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01001110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01111100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01111110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10010110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10011000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10011010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10101100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10110000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10110010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11010010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11010110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11011110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11110100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11110110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11111000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11111010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "86230 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 8470 physical qubits to implement the algorithm logic, and 77760 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "783us 200ns | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (4us 400ns) multiplied by the 178 logical cycles to run the algorithm. If however the duration of a single T factory (here: 46us 800ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "35 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 12\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 35$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "178 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 28 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 22 CCZ and 28 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "178 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 178. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "200 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 22 CCZ and 28 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 200 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 12 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{200\\;\\text{T states} \\cdot 46us 800ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 783us 200ns\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "17 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 200 T states, the 12 copies of the T factory are repeatedly invoked 17 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "8470 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 8470 are the product of the 35 logical qubits after layout and the 242 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "77760 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 6480 physical qubits and we run 12 in parallel, therefore we need $77760 = 6480 \\cdot 12$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "8.03e-8 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 35 logical qubits and the total cycle count 178. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "2.50e-6 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 200. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "11 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.00000008025682182985554)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "242 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "4us 400ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-8 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{11 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 4us 400ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 242. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "6480 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.17e-6. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "9 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "6480 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.17e-6 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 2.50e-6. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 35 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "22 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "28 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "28 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "| Qubit IDs | \r\n", + "0, 1, 2, 3, 4, 5, 6 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|0000000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0000110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101101\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110000\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110011\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110110\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111001\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111010\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111100\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111111\\right\\rangle$ | \r\n", + "$0.1250 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "85746 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 7986 physical qubits to implement the algorithm logic, and 77760 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "413us 600ns | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (4us 400ns) multiplied by the 94 logical cycles to run the algorithm. If however the duration of a single T factory (here: 46us 800ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "33 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 11\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 33$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "94 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 16 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 10 CCZ and 16 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "94 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 94. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "104 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 10 CCZ and 16 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 104 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 12 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{104\\;\\text{T states} \\cdot 46us 800ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 413us 600ns\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "9 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 104 T states, the 12 copies of the T factory are repeatedly invoked 9 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "7986 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 7986 are the product of the 33 logical qubits after layout and the 242 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "77760 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 6480 physical qubits and we run 12 in parallel, therefore we need $77760 = 6480 \\cdot 12$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "1.61e-7 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 33 logical qubits and the total cycle count 94. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "4.81e-6 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 104. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "11 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.00000016118633139909737)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "242 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "4us 400ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-8 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{11 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 4us 400ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 242. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "6480 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.17e-6. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "9 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "6480 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.17e-6 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 4.81e-6. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "11 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 33 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "10 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "16 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "16 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "| Qubit IDs | \r\n", + "0, 1, 2, 3, 4, 5, 6, 7, 8 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "86956 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 9196 physical qubits to implement the algorithm logic, and 77760 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "774us 400ns | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (4us 400ns) multiplied by the 176 logical cycles to run the algorithm. If however the duration of a single T factory (here: 46us 800ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "38 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 13\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 38$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "176 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 32 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 16 CCZ and 32 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "176 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 176. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "192 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 16 CCZ and 32 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 192 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 12 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{192\\;\\text{T states} \\cdot 46us 800ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 774us 400ns\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "16 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 192 T states, the 12 copies of the T factory are repeatedly invoked 16 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "9196 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 9196 are the product of the 38 logical qubits after layout and the 242 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "77760 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 6480 physical qubits and we run 12 in parallel, therefore we need $77760 = 6480 \\cdot 12$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "7.48e-8 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 38 logical qubits and the total cycle count 176. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "2.60e-6 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 192. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "11 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.00000007476076555023923)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "242 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "4us 400ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-8 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{11 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 4us 400ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 242. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "6480 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.17e-6. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "9 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "6480 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.17e-6 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 2.60e-6. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "13 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 38 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "16 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "32 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "32 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "| Qubit IDs | \r\n", + "0, 1, 2, 3, 4, 5, 6, 7 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|00000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00011000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00110000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00110010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00110110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00111000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00111010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00111110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01001110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01101100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01111010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01111100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01111110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10011000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10101100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10110100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11001110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11011000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11100100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11100110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11110100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11111000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11111100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "153186 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 7986 physical qubits to implement the algorithm logic, and 145200 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "1ms 236us 400ns | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (4us 400ns) multiplied by the 281 logical cycles to run the algorithm. If however the duration of a single T factory (here: 57us 200ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "33 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 11\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 33$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "281 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 47 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 31 CCZ and 47 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "281 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 281. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "312 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 31 CCZ and 47 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "15 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 312 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 15 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{312\\;\\text{T states} \\cdot 57us 200ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 1ms 236us 400ns\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "21 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 312 T states, the 15 copies of the T factory are repeatedly invoked 21 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "7986 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 7986 are the product of the 33 logical qubits after layout and the 242 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "145200 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 9680 physical qubits and we run 15 in parallel, therefore we need $145200 = 9680 \\cdot 15$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "5.39e-8 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 33 logical qubits and the total cycle count 281. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "1.60e-6 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 312. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "11 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.00000005391998274560552)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "242 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "4us 400ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-8 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{11 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 4us 400ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 242. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "9680 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "57us 200ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.48e-7. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "11 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "9680 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "57us 200ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.48e-7 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 1.60e-6. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "11 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 33 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "31 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "47 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "47 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "| Qubit IDs | \r\n", + "0, 1, 2, 3, 4, 5, 6, 7 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|00000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00001110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00010110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00011000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00011010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0001111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00100110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00101100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0010111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00110000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00110010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00110100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00111000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0011110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|00111110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0100111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0101110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01011110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01100100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0110110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01110110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|0111100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01111010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01111100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|01111110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10000110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10001100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1000111100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10010010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10010100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10011000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1001110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10011110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10100100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10101000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1010110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10110000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10110110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1011100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|10111110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11000100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100011100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11001000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100101100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1100110100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11001110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11010000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11010110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1101100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11011010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11011100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11011110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11100000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110001100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110010100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11100110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1110100100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11101110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|1111000100000000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11110010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11110100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11110110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11111000\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11111010\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11111100\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|11111110\\right\\rangle$ | \r\n", + "$0.0884 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "129004 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 12844 physical qubits to implement the algorithm logic, and 116160 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "4ms 451us 200ns | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (5us 200ns) multiplied by the 856 logical cycles to run the algorithm. If however the duration of a single T factory (here: 57us 200ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "38 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 13\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 38$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "856 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 166 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 64 CCZ and 166 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "856 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 856. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "920 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 64 CCZ and 166 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 920 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 12 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{920\\;\\text{T states} \\cdot 57us 200ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 4ms 451us 200ns\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "77 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 920 T states, the 12 copies of the T factory are repeatedly invoked 77 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "12844 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 12844 are the product of the 38 logical qubits after layout and the 338 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "116160 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 9680 physical qubits and we run 12 in parallel, therefore we need $116160 = 9680 \\cdot 12$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "1.54e-8 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 38 logical qubits and the total cycle count 856. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "5.43e-7 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 920. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "13 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.000000015371372356123953)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "338 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "5us 200ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-9 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{13 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 5us 200ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 338. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "9680 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "57us 200ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.48e-7. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "11 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "9680 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "57us 200ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.48e-7 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 5.43e-7. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "13 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 38 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "64 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "166 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "166 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "| Qubit IDs | \r\n", + "0, 1, 2, 3, 4, 5, 6, 7, 8 | \r\n", + "||
|---|---|---|---|
| Basis state (bitstring) | \r\n", + "Amplitude | Meas. Pr. | \r\n", + "|
| $\\left|000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000011110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000100110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000101110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000110110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|000111110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001000110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001001110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001010110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001011110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001100110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001101110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001110110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|001111110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010001110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010011110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010100110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010101110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010110110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|010111110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011000110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011001110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011010110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011011110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011100110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101000100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011101110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011110110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111000100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|011111110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100001110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100010110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100011110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100100110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100101110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100110110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|100111110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101000110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101001110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101010110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101011110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101100110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101101110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101110110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111000100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|101111110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110000110000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110001110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110010110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110011110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110100110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110101110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110110110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|110111110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000010000000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111000110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111001110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111010110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011100\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111011110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111100110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101000100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111101110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110000100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111110110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111000100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111010100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111100100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| $\\left|111111110100000000\\right\\rangle$ | \r\n", + "$0.0625 + 0.0000 i$ | \r\n", + " \r\n", + "\r\n",
+ " | \r\n",
+ " | \r\n",
+ " \r\n", + " \r\n", + " | \r\n",
+ " \r\n",
+ " \r\n",
+ "
| Physical qubits | \r\n", + "86230 | \r\n", + "\r\n",
+ " Number of physical qubits \n", + "\r\n", + "\r\n", + " This value represents the total number of physical qubits, which is the sum of 8470 physical qubits to implement the algorithm logic, and 77760 physical qubits to execute the T factories that are responsible to produce the T states that are consumed by the algorithm. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "796us 400ns | \r\n", + "\r\n",
+ " Total runtime \n", + "\r\n", + "\r\n", + " This is a runtime estimate (in nanosecond precision) for the execution time of the algorithm. In general, the execution time corresponds to the duration of one logical cycle (4us 400ns) multiplied by the 181 logical cycles to run the algorithm. If however the duration of a single T factory (here: 46us 800ns) is larger than the algorithm runtime, we extend the number of logical cycles artificially in order to exceed the runtime of a single T factory. \n", + "\r\n", + " | \r\n",
+ "
| Logical algorithmic qubits | \r\n", + "35 | \r\n", + "\r\n",
+ " Number of logical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " Laying out the logical qubits in the presence of nearest-neighbor constraints requires additional logical qubits. In particular, to layout the \\(Q_{\\rm alg} = 12\\) logical qubits in the input algorithm, we require in total $2 \\cdot Q_{\\rm alg} + \\lceil \\sqrt{8 \\cdot Q_{\\rm alg}}\\rceil + 1 = 35$ logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Algorithmic depth | \r\n", + "181 | \r\n", + "\r\n",
+ " Number of logical cycles for the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm using Parallel Synthesis Sequential Pauli Computation (PSSPC), operations are scheduled in terms of multi-qubit Pauli measurements, for which assume an execution time of one logical cycle. Based on the input algorithm, we require one multi-qubit measurement for the 34 single-qubit measurements, the 0 arbitrary single-qubit rotations, and the 0 T gates, three multi-qubit measurements for each of the 15 CCZ and 34 CCiX gates in the input program, as well as No rotations in algorithm multi-qubit measurements for each of the 0 non-Clifford layers in which there is at least one single-qubit rotation with an arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Logical depth | \r\n", + "181 | \r\n", + "\r\n",
+ " Number of logical cycles performed \n", + "\r\n", + "\r\n", + " This number is usually equal to the logical depth of the algorithm, which is 181. However, in the case in which a single T factory is slower than the execution time of the algorithm, we adjust the logical cycle depth to exceed the T factory's execution time. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states | \r\n", + "196 | \r\n", + "\r\n",
+ " Number of T states consumed by the algorithm \n", + "\r\n", + "\r\n", + " To execute the algorithm, we require one T state for each of the 0 T gates, four T states for each of the 15 CCZ and 34 CCiX gates, as well as No rotations in algorithm for each of the 0 single-qubit rotation gates with arbitrary angle rotation. \n", + "\r\n", + " | \r\n",
+ "
| Number of T factories | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of T factories capable of producing the demanded 196 T states during the algorithm's runtime \n", + "\r\n", + "\r\n", + " The total number of T factories 12 that are executed in parallel is computed as \\(\\left\\lceil\\dfrac{196\\;\\text{T states} \\cdot 46us 800ns\\;\\text{T factory duration}}{1\\;\\text{T states per T factory} \\cdot 796us 400ns\\;\\text{algorithm runtime}}\\right\\rceil\\) \n", + "\r\n", + " | \r\n",
+ "
| Number of T factory invocations | \r\n", + "17 | \r\n", + "\r\n",
+ " Number of times all T factories are invoked \n", + "\r\n", + "\r\n", + " In order to prepare the 196 T states, the 12 copies of the T factory are repeatedly invoked 17 times. \n", + "\r\n", + " | \r\n",
+ "
| Physical algorithmic qubits | \r\n", + "8470 | \r\n", + "\r\n",
+ " Number of physical qubits for the algorithm after layout \n", + "\r\n", + "\r\n", + " The 8470 are the product of the 35 logical qubits after layout and the 242 physical qubits that encode a single logical qubit. \n", + "\r\n", + " | \r\n",
+ "
| Physical T factory qubits | \r\n", + "77760 | \r\n", + "\r\n",
+ " Number of physical qubits for the T factories \n", + "\r\n", + "\r\n", + " Each T factory requires 6480 physical qubits and we run 12 in parallel, therefore we need $77760 = 6480 \\cdot 12$ qubits. \n", + "\r\n", + " | \r\n",
+ "
| Required logical qubit error rate | \r\n", + "7.89e-8 | \r\n", + "\r\n",
+ " The minimum logical qubit error rate required to run the algorithm within the error budget \n", + "\r\n", + "\r\n", + " The minimum logical qubit error rate is obtained by dividing the logical error probability 5.00e-4 by the product of 35 logical qubits and the total cycle count 181. \n", + "\r\n", + " | \r\n",
+ "
| Required logical T state error rate | \r\n", + "2.55e-6 | \r\n", + "\r\n",
+ " The minimum T state error rate required for distilled T states \n", + "\r\n", + "\r\n", + " The minimum T state error rate is obtained by dividing the T distillation error probability 5.00e-4 by the total number of T states 196. \n", + "\r\n", + " | \r\n",
+ "
| Number of T states per rotation | \r\n", + "No rotations in algorithm | \r\n", + "\r\n",
+ " Number of T states to implement a rotation with an arbitrary angle \n", + "\r\n", + "\r\n", + " The number of T states to implement a rotation with an arbitrary angle is \\(\\lceil 0.53 \\log_2(0 / 0) + 5.3\\rceil\\) [arXiv:2203.10064]. For simplicity, we use this formula for all single-qubit arbitrary angle rotations, and do not distinguish between best, worst, and average cases. \n", + "\r\n", + " | \r\n",
+ "
| QEC scheme | \r\n", + "surface_code | \r\n", + "\r\n",
+ " Name of QEC scheme \n", + "\r\n", + "\r\n", + " You can load pre-defined QEC schemes by using the name | \r\n",
+ "
| Code distance | \r\n", + "11 | \r\n", + "\r\n",
+ " Required code distance for error correction \n", + "\r\n", + "\r\n", + " The code distance is the smallest odd integer greater or equal to \\(\\dfrac{2\\log(0.03 / 0.00000007892659826361484)}{\\log(0.01/0.001)} - 1\\) \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "242 | \r\n", + "\r\n",
+ " Number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " The number of physical qubits per logical qubit are evaluated using the formula 2 * codeDistance * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time | \r\n", + "4us 400ns | \r\n", + "\r\n",
+ " Duration of a logical cycle in nanoseconds \n", + "\r\n", + "\r\n", + " The runtime of one logical cycle in nanoseconds is evaluated using the formula (4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance that can be user-specified. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubit error rate | \r\n", + "3.00e-8 | \r\n", + "\r\n",
+ " Logical qubit error rate \n", + "\r\n", + "\r\n", + " The logical qubit error rate is computed as $0.03 \\cdot \\left(\\dfrac{0.001}{0.01}\\right)^\\frac{11 + 1}{2}$ \n", + "\r\n", + " | \r\n",
+ "
| Crossing prefactor | \r\n", + "0.03 | \r\n", + "\r\n",
+ " Crossing prefactor used in QEC scheme \n", + "\r\n", + "\r\n", + " The crossing prefactor is usually extracted numerically from simulations when fitting an exponential curve to model the relationship between logical and physical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Error correction threshold | \r\n", + "0.01 | \r\n", + "\r\n",
+ " Error correction threshold used in QEC scheme \n", + "\r\n", + "\r\n", + " The error correction threshold is the physical error rate below which the error rate of the logical qubit is less than the error rate of the physical qubit that constitute it. This value is usually extracted numerically from simulations of the logical error rate. \n", + "\r\n", + " | \r\n",
+ "
| Logical cycle time formula | \r\n", + "(4 * twoQubitGateTime + 2 * oneQubitMeasurementTime) * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute logical cycle time \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the logical cycle time 4us 400ns. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits formula | \r\n", + "2 * codeDistance * codeDistance | \r\n", + "\r\n",
+ " QEC scheme formula used to compute number of physical qubits per logical qubit \n", + "\r\n", + "\r\n", + " This is the formula that is used to compute the number of physical qubits per logical qubits 242. \n", + "\r\n", + " | \r\n",
+ "
| Physical qubits | \r\n", + "6480 | \r\n", + "\r\n",
+ " Number of physical qubits for a single T factory \n", + "\r\n", + "\r\n", + " This corresponds to the maximum number of physical qubits over all rounds of T distillation units in a T factory. A round of distillation contains of multiple copies of distillation units to achieve the required success probability of producing a T state with the expected logical T state error rate. \n", + "\r\n", + " | \r\n",
+ "
| Runtime | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " Runtime of a single T factory \n", + "\r\n", + "\r\n", + " The runtime of a single T factory is the accumulated runtime of executing each round in a T factory. \n", + "\r\n", + " | \r\n",
+ "
| Number of output T states per run | \r\n", + "1 | \r\n", + "\r\n",
+ " Number of output T states produced in a single run of T factory \n", + "\r\n", + "\r\n", + " The T factory takes as input 30 noisy physical T states with an error rate of 0.001 and produces 1 T states with an error rate of 2.17e-6. \n", + "\r\n", + " | \r\n",
+ "
| Number of input T states per run | \r\n", + "30 | \r\n", + "\r\n",
+ " Number of physical input T states consumed in a single run of a T factory \n", + "\r\n", + "\r\n", + " This value includes the physical input T states of all copies of the distillation unit in the first round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation rounds | \r\n", + "1 | \r\n", + "\r\n",
+ " The number of distillation rounds \n", + "\r\n", + "\r\n", + " This is the number of distillation rounds. In each round one or multiple copies of some distillation unit is executed. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units per round | \r\n", + "2 | \r\n", + "\r\n",
+ " The number of units in each round of distillation \n", + "\r\n", + "\r\n", + " This is the number of copies for the distillation units per round. \n", + "\r\n", + " | \r\n",
+ "
| Distillation units | \r\n", + "15-to-1 space efficient logical | \r\n", + "\r\n",
+ " The types of distillation units \n", + "\r\n", + "\r\n", + " These are the types of distillation units that are executed in each round. The units can be either physical or logical, depending on what type of qubit they are operating. Space-efficient units require fewer qubits for the cost of longer runtime compared to Reed-Muller preparation units. \n", + "\r\n", + " | \r\n",
+ "
| Distillation code distances | \r\n", + "9 | \r\n", + "\r\n",
+ " The code distance in each round of distillation \n", + "\r\n", + "\r\n", + " This is the code distance used for the units in each round. If the code distance is 1, then the distillation unit operates on physical qubits instead of error-corrected logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| Number of physical qubits per round | \r\n", + "6480 | \r\n", + "\r\n",
+ " The number of physical qubits used in each round of distillation \n", + "\r\n", + "\r\n", + " The maximum number of physical qubits over all rounds is the number of physical qubits for the T factory, since qubits are reused by different rounds. \n", + "\r\n", + " | \r\n",
+ "
| Runtime per round | \r\n", + "46us 800ns | \r\n", + "\r\n",
+ " The runtime of each distillation round \n", + "\r\n", + "\r\n", + " The runtime of the T factory is the sum of the runtimes in all rounds. \n", + "\r\n", + " | \r\n",
+ "
| Logical T state error rate | \r\n", + "2.17e-6 | \r\n", + "\r\n",
+ " Logical T state error rate \n", + "\r\n", + "\r\n", + " This is the logical T state error rate achieved by the T factory which is equal or smaller than the required error rate 2.55e-6. \n", + "\r\n", + " | \r\n",
+ "
| Logical qubits (pre-layout) | \r\n", + "12 | \r\n", + "\r\n",
+ " Number of logical qubits in the input quantum program \n", + "\r\n", + "\r\n", + " We determine 35 from this number by assuming to align them in a 2D grid. Auxiliary qubits are added to allow for sufficient space to execute multi-qubit Pauli measurements on all or a subset of the logical qubits. \n", + "\r\n", + " | \r\n",
+ "
| T gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of T gates in the input quantum program \n", + "\r\n", + "\r\n", + " This includes all T gates and adjoint T gates, but not T gates used to implement rotation gates with arbitrary angle, CCZ gates, or CCiX gates. \n", + "\r\n", + " | \r\n",
+ "
| Rotation gates | \r\n", + "0 | \r\n", + "\r\n",
+ " Number of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all rotation gates. If an angle corresponds to a Pauli, Clifford, or T gate, it is not accounted for in this number. \n", + "\r\n", + " | \r\n",
+ "
| Rotation depth | \r\n", + "0 | \r\n", + "\r\n",
+ " Depth of rotation gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of all non-Clifford layers that include at least one single-qubit rotation gate with an arbitrary angle. \n", + "\r\n", + " | \r\n",
+ "
| CCZ gates | \r\n", + "15 | \r\n", + "\r\n",
+ " Number of CCZ-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCZ gates. \n", + "\r\n", + " | \r\n",
+ "
| CCiX gates | \r\n", + "34 | \r\n", + "\r\n",
+ " Number of CCiX-gates in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of CCiX gates, which applies \\(-iX\\) controlled on two control qubits [1212.5069]. \n", + "\r\n", + " | \r\n",
+ "
| Measurement operations | \r\n", + "34 | \r\n", + "\r\n",
+ " Number of single qubit measurements in the input quantum program \n", + "\r\n", + "\r\n", + " This is the number of single qubit measurements in Pauli basis that are used in the input program. Note that all measurements are counted, however, the measurement result is is determined randomly (with a fixed seed) to be 0 or 1 with a probability of 50%. \n", + "\r\n", + " | \r\n",
+ "
| Total error budget | \r\n", + "1.00e-3 | \r\n", + "\r\n",
+ " Total error budget for the algorithm \n", + "\r\n", + "\r\n", + " The total error budget sets the overall allowed error for the algorithm, i.e., the number of times it is allowed to fail. Its value must be between 0 and 1 and the default value is 0.001, which corresponds to 0.1%, and means that the algorithm is allowed to fail once in 1000 executions. This parameter is highly application specific. For example, if one is running Shor's algorithm for factoring integers, a large value for the error budget may be tolerated as one can check that the output are indeed the prime factors of the input. On the other hand, a much smaller error budget may be needed for an algorithm solving a problem with a solution which cannot be efficiently verified. This budget \\(\\epsilon = \\epsilon_{\\log} + \\epsilon_{\\rm dis} + \\epsilon_{\\rm syn}\\) is uniformly distributed and applies to errors \\(\\epsilon_{\\log}\\) to implement logical qubits, an error budget \\(\\epsilon_{\\rm dis}\\) to produce T states through distillation, and an error budget \\(\\epsilon_{\\rm syn}\\) to synthesize rotation gates with arbitrary angles. Note that for distillation and rotation synthesis, the respective error budgets \\(\\epsilon_{\\rm dis}\\) and \\(\\epsilon_{\\rm syn}\\) are uniformly distributed among all T states and all rotation gates, respectively. If there are no rotation gates in the input algorithm, the error budget is uniformly distributed to logical errors and T state errors. \n", + "\r\n", + " | \r\n",
+ "
| Logical error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one logical error \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| T distillation error probability | \r\n", + "5.00e-4 | \r\n", + "\r\n",
+ " Probability of at least one faulty T distillation \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3 if the input algorithm contains rotation with gates with arbitrary angles, or one half of it, otherwise. \n", + "\r\n", + " | \r\n",
+ "
| Rotation synthesis error probability | \r\n", + "0.00e0 | \r\n", + "\r\n",
+ " Probability of at least one failed rotation synthesis \n", + "\r\n", + "\r\n", + " This is one third of the total error budget 1.00e-3. \n", + "\r\n", + " | \r\n",
+ "
| Qubit name | \r\n", + "qubit_gate_ns_e3 | \r\n", + "\r\n",
+ " Some descriptive name for the qubit model \n", + "\r\n", + "\r\n", + " You can load pre-defined qubit parameters by using the names | \r\n",
+ "
| Instruction set | \r\n", + "GateBased | \r\n", + "\r\n",
+ " Underlying qubit technology (gate-based or Majorana) \n", + "\r\n", + "\r\n", + " When modeling the physical qubit abstractions, we distinguish between two different physical instruction sets that are used to operate the qubits. The physical instruction set can be either gate-based or Majorana. A gate-based instruction set provides single-qubit measurement, single-qubit gates (incl. T gates), and two-qubit gates. A Majorana instruction set provides a physical T gate, single-qubit measurement and two-qubit joint measurement operations. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement time | \r\n", + "100 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit measurement (t_meas) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit measurement in the Pauli basis. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for single-qubit gate (t_gate) in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a single-qubit Clifford operation, e.g., Hadamard or Phase gates. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for two-qubit gate in ns \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to perform a two-qubit Clifford operation, e.g., a CNOT or CZ gate. \n", + "\r\n", + " | \r\n",
+ "
| T gate time | \r\n", + "50 ns | \r\n", + "\r\n",
+ " Operation time for a T gate \n", + "\r\n", + "\r\n", + " This is the operation time in nanoseconds to execute a T gate. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit measurement error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit measurement \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit measurement in the Pauli basis may fail. \n", + "\r\n", + " | \r\n",
+ "
| Single-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for single-qubit Clifford gate (p) \n", + "\r\n", + "\r\n", + " This is the probability in which a single-qubit Clifford operation, e.g., Hadamard or Phase gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| Two-qubit error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate for two-qubit Clifford gate \n", + "\r\n", + "\r\n", + " This is the probability in which a two-qubit Clifford operation, e.g., CNOT or CZ gates, may fail. \n", + "\r\n", + " | \r\n",
+ "
| T gate error rate | \r\n", + "0.001 | \r\n", + "\r\n",
+ " Error rate to prepare single-qubit T state or apply a T gate (p_T) \n", + "\r\n", + "\r\n", + " This is the probability in which executing a single T gate may fail. \n", + "\r\n", + " | \r\n",
+ "
More details on the following lists of assumptions can be found in the paper Accessing requirements for scaling quantum computers and their applications.
\n", + "Uniform independent physical noise. We assume that the noise on physical qubits and physical qubit operations is the standard circuit noise model. In particular we assume error events at different space-time locations are independent and that error rates are uniform across the system in time and space.
\n", + "Efficient classical computation. We assume that classical overhead (compilation, control, feedback, readout, decoding, etc.) does not dominate the overall cost of implementing the full quantum algorithm.
\n", + "Extraction circuits for planar quantum ISA. We assume that stabilizer extraction circuits with similar depth and error correction performance to those for standard surface and Hastings-Haah code patches can be constructed to implement all operations of the planar quantum ISA (instruction set architecture).
\n", + "Uniform independent logical noise. We assume that the error rate of a logical operation is approximately equal to its space-time volume (the number of tiles multiplied by the number of logical time steps) multiplied by the error rate of a logical qubit in a standard one-tile patch in one logical time step.
\n", + "Negligible Clifford costs for synthesis. We assume that the space overhead for synthesis and space and time overhead for transport of magic states within magic state factories and to synthesis qubits are all negligible.
\n", + "Smooth magic state consumption rate. We assume that the rate of T state consumption throughout the compiled algorithm is almost constant, or can be made almost constant without significantly increasing the number of logical time steps for the algorithm.
\n", + "