Fix phase management in StabilizerState.expectation_value

[ShellyGarion]

Summary

close #7441

Fix a bug in StabilizerState.expectation_value() method when the Pauli operator has a non-trivial phase.

Details and comments

In addition, verify that the operator in indeed a Pauli, and extend the tests accordingly.

[coveralls]

Pull Request Test Coverage Report for Build 1858344895

• 7 of 8 (87.5%) changed or added relevant lines in 1 file are covered.
• 3 unchanged lines in 1 file lost coverage.
• Overall coverage decreased (-0.006%) to 83.323%

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Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
qiskit/quantum_info/states/stabilizerstate.py	7	8	87.5%

Files with Coverage Reduction	New Missed Lines	%
qiskit/pulse/library/waveform.py	3	89.36%

Totals
Change from base Build 1855099047:	-0.006%
Covered Lines:	52203
Relevant Lines:	62651

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💛 - Coveralls

[ikkoham]

LGTM. Let me make just one issue.

[ikkoham]

If we extend this to SparsePauliOp, we can use from_operator to apply it to the entire BaseOperator. However, I think this is beyond the scope of the bug fix and should be treated as a separate issue.

[ShellyGarion]

I agree that we should open a separate PR to extend this to SparsePauliOp

[ShellyGarion]

I can start such a PR after this PR will be merged

[jakelishman]

In principle the change here looks straightforwards, but I don't actually know the maths, and there's an unchanged line that feels wrong. If it's actually correct, I'm happy to accept if we can beef up the new test case a little further.

[ikkoham]

Thank you for adding tests. I think it's more robust now. Maybe there are some things that could be improved that this PR haven't changed, but I think this is fine.
@jakelishman What do you think?

[1ucian0]

I just changed the tests to follow the ddt patter. For the rest, I trust @ikkoham with the math. Let's get it merged.

[jakelishman]

Sorry that I took so long to come back to this.

The tests look much more complete now, thanks - I've got way more confidence that we're doing the right thing in a lot more circumstances, even without in-depth knowledge of the algorithm.

I just have one outstanding comment/question in the algorithm. I'll approve now to save time in case the comment's way off base, but if not, it might be good just to put a shade more explanation at the bit I mentioned to help future reviewers (like me) who aren't 100% certain about the formalism.

[ShellyGarion]

@jakelishman - here is the algorithm of the expectation value calculation that we got from Sergey and implemented here and in Aer's stabilizer simulator.

Suppose one needs to compute <psi|P|psi> where P is a Pauli operator (with no phase) and psi is a stabilizer state with stabilizer generators S_1,...,S_n and destabilizer generators D_1,...,D_n.

If P anti-commutes with some stabilizer S_j then <psi|P|psi>=0.

Otherwise, P or -P is an element of the stabilizer group and thus P = (-1)^a S_1^{b_1} S_2^{b_2} ... S_n^{b_n}
for some unknown coefficients a,b_1,...,b_n \in {0,1}.
Note that <psi|P|psi> = (-1)^a since stabilizers act trivially on psi. Thus it suffices to compute a.

Note that b_j=1 iff P anti-commutes with the destabilizer D_j.
Thus we can peel off all stabilizers from P as follows:
for j = 1 to n
if P anti-commutes with D_j then multiply P by S_j
end
Now P = (-1)^a times the identity operator. The desired expected value is (-1)^a.

[jakelishman]

Thanks, I think I get it all now. I was getting all confused because we were calculating with a counter on the symplectic phase, but at the end we were only either leaving or negating the return value. The idea that it needed to remain real was fine for my intuition, but while I thought that all values of phase (1, -1, i, -i) were possible, I couldn't understand why only one would behave differently to the others. Since i and -i aren't actually possible, I have no problems.