Add general Measurement

[Mayank447]

Currently qutip-qip only supports measurement in Pauli Z i.e. the standard basis. Expanding this to a general operator basis would be helpful as several algorithms/protocols measure in other basis e.g. Pauli X while other like OTOCs (Out of Time Ordered Correlators) inherently rely randomised measurements.

Ideally along side general measurement we should add easier methods to QubitCircuit e.g. qc.measure_all() and also expand (or refactor) the Measurement class as target, cbits are being moved to MeasurementInstruction in gate refactor. There are couple of more things I have in mind which can be added alongside controlled measurements, operations on cbits and POVM measurements.

[mudit06mah]

Hi @Mayank447
Are you working on this or is it okay for me to? Since this is now blocking the Ops feature issue i can work on this first.

[Mayank447]

Sure go ahead, I haven't started work on this. Also before starting with code implementation, open a discussion tab for this and propose your implementation design there.

[mudit06mah]

Further Discussion in #373

[mudit06mah]

Checklist:

• Shift classical_store to Qubit Circuit
• Remove targets and classical_store from measurement.py
• Expand Measurement Class: num_qubits, get_qobj(), apply(), get_measurement_ops()
• Implement operator subclasses (Mx, My, Mz)
• Update simulator to use arbitrary bases and map multi-qubit results
• Add measure_all() method to circuit.py
• Add basis conversion (Mx, My -> Mz) to to_qasm() in instructions.py

[Mayank447]

Adding this as a future note:

Naimark's (Neumark's) Dilation Theorem
The physical realization of a POVM is explained by Naimark's Dilation Theorem. The theorem states that any generalized POVM on a principal system can be physically realized by:

1. Attaching an ancilla system.
2. Applying a unitary transformation to the joint system (entangling the principal system and ancilla).
3. Performing a standard orthogonal projective measurement (PVM) on the larger joint system.

This is needed for simulating POVMs on the Pulse Simulator, though adding projective measurement to Pulse Simulator is pending and will need quite a bit of work. But once Projective Measurements are implemented for Pulse Simulator, this implies POVMs are automatically as well.