constructing cliffords from linear functions and permutation gates

[alexanderivrii]

Summary

Adding Clifford.from_linear_function and Clifford.from_permutation methods, and allowing to construct a Clifford either directly from a linear function or a permutation, or from a quantum circuit containing such gates.

Details and comments

Previously, Cliffords could be constructed from linear functions (or from circuits containing linear functions), however this required to first synthesize each linear function into a circuit consisting of CX and SWAP gates and then to create a Clifford from this circuit. This is obviously slower than the direct construction: when a linear function is represented by a nxn binary invertible matrix A, then the corresponding Clifford has the symplectic matrix [[A^{t}, 0], [0, A^{-1}]].

Just for fun, here are some numbers showing the times for the old and the new constructions, summed over 10 different linear functions with the right number of qubits:

num_qubits   old time      new time
  5          0.0150          0.0020
 10          0.0614          0.0030
 50          2.1066          0.0259
100          9.0950          0.1092
250          66.7331         0.7266

Both are very fast when the number of qubits is in the small-medium range, but after that the time to synthesize a linear function becomes noticeable.

Previously, Cliffords could not be constructed from permutation gates (i.e. from PermutationGate objects), though they could be constructed from Permutation circuits, which again required to synthesize them first. Again, a direct and much easier construction is available (not to mention that permutations are special cases of linear functions).

[qiskit-bot]

One or more of the the following people are requested to review this:

• @Qiskit/terra-core
• @ikkoham

[coveralls]

Pull Request Test Coverage Report for Build 5846291654

• 42 of 42 (100.0%) changed or added relevant lines in 2 files are covered.
• 22 unchanged lines in 4 files lost coverage.
• Overall coverage decreased (-0.01%) to 87.246%

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Files with Coverage Reduction	New Missed Lines	%
crates/qasm2/src/expr.rs	1	93.76%
qiskit/pulse/library/waveform.py	3	93.75%
crates/qasm2/src/lex.rs	6	90.38%
crates/qasm2/src/parse.rs	12	96.67%

Totals
Change from base Build 5837476817:	-0.01%
Covered Lines:	74322
Relevant Lines:	85187

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💛 - Coveralls

[ShellyGarion]

This PR is well-written and will be a good contribution to Qiskit.

I only have a minor comment, in this PR it is worth to update the test from PR #10053:
https://github.com/Qiskit/qiskit-terra/blob/06416d7ee75500c50cfddc67574421cf73913f8a/test/python/circuit/library/test_linear_function.py#L489
(I checked the code and it should work now)

[alexanderivrii]

Thanks @ShellyGarion, good catch! I have updated the test in 7cb3a4f.

[ikkoham]

LGTM. TBH, I'm a little concerned about the cyclic import, but that's due to the original design.