English suggestions to Shor Factoring

Author: AnnePicus

algorithms/algebraic/shor/shor.ipynb

@@ -19,30 +19,28 @@
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-    "## Introduction\n",
+    "The integer factorization problem [[1](#IntegerFactor)] is a famous problem in number theory: given a composite number $N$, find its prime factors. The importance of the problem stems from there being no known efficient (polynomial-time, in the number of bits needed to represent $N$) classical algorithm, and much of modern-day cryptography relies on this fact. In 1994, Peter Shor came up with an efficient _quantum_ algorithm for the problem [[2](#Shor94)], providing one of the first concrete pieces of evidence for the power of quantum computers.\n",
     "\n",
-    "The integer factorization problem [[1](#IntegerFactor)] is a famous problem in number theory: given a number $N$ which is composite, find its prime factors. The importance of the problem stems from the fact that no efficient (polynomial-time, in the number of bits needed to represent $N$) classical algorithm is known for it to this day, and much of modern day cryptography relies on this fact. In 1994, Peter Shor came up with an efficient _quantum_ algorithm for the problem [[2](#Shor94)] - providing one of the first concrete pieces of evidence for the power of quantum computers.\n",
+    "## Shor's Algorithm\n",
     "\n",
-    "### Shor's Algorithm\n",
+    "Shor's algorithm consists of a classical part and a quantum subroutine. The steps of the algorithm for factoring an input number $N$, summarized from [[3](#ShorSteps)]:\n",
     "\n",
-    "Shor's algorithm consists of a classical part and a quantum subroutine. The steps of the algorithm for factoring an input number $N$, summarized from [[3](#ShorSteps)], are as follows:\n",
-    "\n",
-    "1. Pick a random number $1 < a < N$ that is co-prime with $N$. Co-primality can be checked by computing the GCD (greatest common divisor) of $a$ and $N$ - if it is 1 then we have found a co-prime $a$, otherwise we have found a non-trivial factor of $N$ and we are done.\n",
+    "1. Pick a random number $1 < a < N$ that is co-prime with $N$. Co-primality can be checked by computing the GCD (greatest common divisor) of $a$ and $N$. If it is 1 then we have a co-prime $a$, otherwise we have a non-trivial factor of $N$ and we are done.\n",
     "2. Find the period $r$ of the following function, using the quantum period finding algorithm (described in [[4](#PeriodFinding)]): $$f(x) = a^x \\mod N$$\n",
     "3. If $r$ is odd or $a^{r/2} = -1 \\mod N$, return to step 1 (this event can be shown to happen with probability at most $1/2$).\n",
     "4. Otherwise, $\\gcd(a^{r/2} \\pm 1, N)$ are both factors of $N$, and computing one of them yields the required result.\n",
     "\n",
-    "In this demo, we will factor the number $N=15$ using Shor's algorithm, by applying the quantum subroutine (step 2) with $a=7$. This particular $a$ is chosen since it is co-prime with 15 and satisfies the conditions of step 3, providing us with a high probability of finding a factor of $N$.\n"
+    "This demo factors the number $N=15$ using Shor's algorithm by applying the quantum subroutine (step 2) with $a=7$. We choose this particular $a$ since it is co-prime with 15 and satisfies the conditions of step 3, providing us with a high probability of finding a factor of $N$.\n"
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-    "## Building the quantum period finding circuit\n",
+    "## Building the Quantum Period Finding Circuit\n",
     "\n",
-    "We begin by declaring the number of qubits in the upper (counting) register the quantum subroutine uses. In our case, $N = 15$, and according to the algorithm the upper register must contain $q = \\log(Q)$ qubits for $Q$ such that $N^2 \\le Q < 2N^2$, namely $225 < Q < 450$, and therefore $q = 8$. In addition, the second register should be large enough to encode 15, hence:"
+    "We begin by declaring the number of qubits in the upper (counting) register that the quantum subroutine uses. In our case, $N = 15$, and according to the algorithm the upper register must contain $q = \\log(Q)$ qubits for $Q$ such that $N^2 \\le Q < 2N^2$; namely, $225 < Q < 450$, and therefore $q = 8$. In addition, the second register should be large enough to encode 15, hence:"
    ]
   },
   {
@@ -67,11 +65,11 @@
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-    "We will implement a Phase Estimation [[5](#PhaseEstimation)] circuit. Each element in the circuit is a controlled operation of: $$|x\\rangle \\rightarrow |x\\cdot a^{2^i}\\mod N \\rangle $$ where $a < N$ is a number such that $\\gcd(a, N)=1$. For this demonstration we picked $a=7$. $i$ is the index of the control qubit, located in the upper register.\n",
+    "We implement a Phase Estimation [[5](#PhaseEstimation)] circuit. Each element in the circuit is a controlled operation of $$|x\\rangle \\rightarrow |x\\cdot a^{2^i}\\mod N \\rangle $$ where $a < N$ is a number such that $\\gcd(a, N)=1$. For this demonstration we pick $a=7$. $i$ is the index of the control qubit, located in the upper register.\n",
     "\n",
-    "It is quiet involved to implement these unitaries, so for this demo we will make a shortcut, and compute exactly the unitary matrix that implements the computation (which in the general case is not applicable as this pre-processing is exponential). We will do so by calculating the modular-multiplication by $a$ matrix, then using its powers.\n",
+    "It is quite involved to implement these unitaries, so for this demo we take a shortcut and compute the exact unitary matrix that implements the computation (which in general is not applicable as this preprocessing is exponential). We do so by calculating the modular-multiplication by $a$ matrix and then using its powers.\n",
     "\n",
-    "The function `unitary` is used for decomposing the unitary matrix into quantum gates."
+    "The `unitary` function is used for decomposing the unitary matrix into quantum gates."
    ]
   },
   {
@@ -122,7 +120,7 @@
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-    "### Building the complete circuit"
+    "### Complete Circuit"
    ]
   },
   {
@@ -138,8 +136,8 @@
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-    "We then apply the second layer of the circuit, which consists of the controlled $U^{2^i}$ gates. \n",
-    "Lastly, we apply an inverse QFT on the counting register, to get the period."
+    "We then apply the second layer of the circuit, consisting of the controlled $U^{2^i}$ gates. \n",
+    "Lastly, we apply an inverse QFT on the counting register to get the period."
    ]
   },
   {
@@ -156,15 +154,15 @@
     "    qv_counting: Output[QArray[QBit, num_counting_qubits]],\n",
     "    qv_auxilliary: Output[QArray[QBit, num_auxilliary_qubits]],\n",
     ") -> None:\n",
-    "    # start with a hadamard transform in the counting register\n",
+    "    # start with a Hadamard transform in the counting register\n",
     "    allocate(num_counting_qubits, qv_counting)\n",
     "    hadamard_transform(qv_counting)\n",
     "\n",
     "    # Prepare the |1> state on the lower register\n",
     "    allocate(num_auxilliary_qubits, qv_auxilliary)\n",
     "    X(qv_auxilliary[0])\n",
     "\n",
-    "    # Apply the contolled modular-exponentiations using each of the counting qubits\n",
+    "    # Apply the controlled modular-exponentiations using each of the counting qubits\n",
     "    repeat(\n",
     "        count=num_auxilliary_qubits,\n",
     "        iteration=lambda index: control(\n",
@@ -182,10 +180,10 @@
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-    "### Quantum entry point\n",
-    "In order to synthesize the circuit, we define a quantum `main` function. As are we only interested in the output of the counting register, we only define it in the signature of the function.\n",
+    "### Quantum Entry Point\n",
+    "To synthesize the circuit, we define a quantum `main` function. As we are only interested in the output of the counting register, we only define it in the signature of the function.\n",
     "\n",
-    "Next, we translate it to qmod using the `create_model`."
+    "Next, we translate it to Qmod using `create_model`."
    ]
   },
   {
@@ -211,7 +209,7 @@
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-    "We now send the model to the synthesis engine, taking a few seconds:"
+    "We now send the model to the synthesis engine, which may take a few seconds:"
    ]
   },
   {
@@ -245,9 +243,7 @@
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@@ -261,9 +257,9 @@
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-    "## Executing the circuit\n",
+    "## Executing the Circuit\n",
     "\n",
-    "Now, we turn to executing the circuit above, using the simulator:"
+    "Now, we execute the circuit above, using the simulator:"
    ]
   },
   {
@@ -342,13 +338,13 @@
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-    "We obtained 4 results $y$ from the circuit, each with probability roughly $1/4$: $0, 64, 128$ and $192$. Dividing by $Q = 256$ we obtain 4 reduced fractions: $0, 1/4, 1/2$ and $3/4$, with two of them having the correct period $r=4$ in the denominator. With this period, we can compute the factors of $N = 15$: $\\gcd(a^{r/2} \\pm 1, N) = \\gcd(7^2 \\pm 1, 15) = 3, 5$.\n",
+    "We obtained four $y$ results from the circuit, each with a rough probability of $1/4$: $0, 64, 128$, and $192$. By dividing by $Q = 256$ we obtain four reduced fractions: $0, 1/4, 1/2$, and $3/4$, with two of them having the correct period $r=4$ in the denominator. With this period, we can compute the factors of $N = 15$: $\\gcd(a^{r/2} \\pm 1, N) = \\gcd(7^2 \\pm 1, 15) = 3, 5$.\n",
     "\n",
     "## References\n",
     "\n",
     "<a id='IntegerFactor'>[1]</a>: [Integer Factorization (Wikipedia)](https://en.wikipedia.org/wiki/Integer_factorization)\n",
     "\n",
-    "<a id='Shor94'>[2]</a>: [Shor, Peter W. \"Algorithms for quantum computation: discrete logarithms and factoring.\" Proceedings 35th annual symposium on foundations of computer science. Ieee, 1994.](https://ieeexplore.ieee.org/abstract/document/365700)\n",
+    "<a id='Shor94'>[2]</a>: [Shor, Peter W. \"Algorithms for quantum computation: Discrete logarithms and factoring.\" Proceedings 35th annual symposium on foundations of computer science. IEEE, 1994.](https://ieeexplore.ieee.org/abstract/document/365700)\n",
     "\n",
     "<a id='ShorSteps'>[3]</a>: [Shor's Algorithm Procedure (Wikipedia)](https://en.wikipedia.org/wiki/Shor%27s_algorithm#Procedure)\n",
     "\n",