Add circuit generation from set of stabilizers

[Randl]

• I have added the tests to cover my changes.
• I have updated the documentation accordingly.
• I have read the CONTRIBUTING document.

Summary

Add stabilizer_to_circuit function that given a set of stabilizers returns a (Clifford) circuit whose output (on zero input) is stabilized by this set.

Closes #11478

Details and comments

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[ShellyGarion]

@Randl - thank you very much for your contribution to Qiskit! Here are a few initial comments:

• Since the main function stabilizer_to_circuit is in fact a synthesis function, then it seems that it would better fit into the synthesis library here: https://docs.quantum.ibm.com/api/qiskit/synthesis#stabilizer-state-synthesis
  To fit with the naming convention, perhaps you can call it: synth_stabilizer_from_generators ?

• All other helper functions should be internal, and so their names should start with an underscore, e.g. add_sign should be _add_sign.

• In addition, perhaps we can also add it as a method to the StabilizerState class? we can call it from_stabilizers_to_circuit ?
  Similar to what we did here:

  qiskit/qiskit/quantum_info/operators/symplectic/clifford.py

  Line 635 in 733e2a3

  def to_circuit(self):

• How do you handle signs? Generally, the stabilizers could also have +/- signs (these are the phases of the Clifford tableau), which translate to the corresponding Pauli gates when you construct the circuit. I didn't see tests for this.

• I would also be happy to see some pseudo-random tests, that check that you get back the same stabilizers, like:

cliff = random_clifford(num_qubits, seed=1234)
stabs = cliff.to_labels(mode="S")
qc = stabilizer_to_circuit(stabs)
cliff1 = Clifford(qc)
cliff1.to_labels(mode="S") # should be the same as stabs up to a permutation

See similar pseudo-random tests in: https://github.com/Qiskit/qiskit/blob/main/test/python/synthesis/test_stabilizer_synthesis.py

• The docstrings in your function are not standard in Qiskit. Could you please update them to Qiskit's convention?
  See other files in the synthesis library: https://github.com/Qiskit/qiskit/tree/main/qiskit/synthesis

• Also the tests should be moved to: https://github.com/Qiskit/qiskit/tree/main/test/python/synthesis

[Randl]

@ShellyGarion
I think I've addressed your comments

[ShellyGarion]

Hi @Randl - I very much like your PR and think it will be an important contribution to Qiskit.
I have a few additional minor comments.
I would also like @alexanderivrii to review this PR.

[alexanderivrii]

Overall, this looks good.

I have left inline a few minor nitpicks about formatting and possibility for code reuse, but nothing blocking.

A quick question about the implementation: is there an advantage to working with the string representation of stabilizers (such as "+XIZI") vs. converting them to Paulis and working with the underlying nd.arrays?

Also a question about the scalability of the approach: how much time does this take to synthesize stabilizer states for 100 or so qubits?

[alexanderivrii]

We usually use Returns: instead of Return: (though I do see Return: in multiple places in the code as well). Let me address this to a higher autority, @jakelishman.

[Randl]

@alexanderivrii you're correct about using Pauli; it makes code cleaner. I've rewritten the algorithm removing string usage.
As for runtime, the algorithm is quadratic, but the constant is not too small. Locally, on M1 MacBook, it takes around 15-20 seconds to run a single random test (generate random Clifford + run the algorithm + check equivalence) with 100 qubits.
I've tried to address other comments, though some are irrelevant after rewrite.

[alexanderivrii]

Thank you for quickly addressing the previous comments (and indeed the code is now much easier to read). LGTM!