Adding linear synthesis algorithm for LNN

[ShellyGarion]

Summary

Adding a synthesis algorithm for linear reversible circuits (containing only CX gates) for linear-nearest-neighbor connectivity.
See #9036

Co-authored by Ben Zindorf.

Details and comments

According to the paper: https://arxiv.org/abs/quant-ph/0701194
The depth of an n-qubit linear reversible circuit is bounded by 5n.

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[coveralls]

Pull Request Test Coverage Report for Build 3575999269

• 123 of 125 (98.4%) changed or added relevant lines in 4 files are covered.
• 5 unchanged lines in 2 files lost coverage.
• Overall coverage increased (+0.06%) to 84.599%

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Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
qiskit/synthesis/linear/linear_depth_lnn.py	117	119	98.32%

Files with Coverage Reduction	New Missed Lines	%
src/sabre_swap/layer.rs	2	98.95%
qiskit/pulse/library/waveform.py	3	91.49%

Totals
Change from base Build 3565432800:	0.06%
Covered Lines:	62983
Relevant Lines:	74449

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💛 - Coveralls

[alexanderivrii]

Thanks @ShellyGarion! Overall, this looks good to (and I have the benefit of knowing that this code works correctly in practice). I have left a few minor comments, nothing too deep.

[alexanderivrii]

Thanks a lot for the improvements! I have left a few more requests for minor changes, otherwise this looks very good.

My main problem when looking at this code (which has nothing to do with the code quality itself) it that the proof for lemma 7.3 in the paper is quite involved and does not fully describe how to implement it efficiently. I now see that Ben came up with the idea based on using the inverse matrix to get an efficient implementation.

Other than that the code follows the paper quite closely, and contains very helpful comments.

[alexanderivrii]

Thanks @ShellyGarion, LGTM!

[mtreinish]

Is this including in the docs tree anywhere? We should make sure that the new function we're advertising in the release notes has its documentation published.

[ShellyGarion]

I was not sure if we want to have this standalone function in the API docs, or allow to access it only via the HLS synthesis plugin

[ShellyGarion]

But if we add this algorithm to docs, then we should also add Patel-Markov-Hayes as well

[alexanderivrii]

Thanks, @ShellyGarion, looks good to me (including downloading documentation artifacts to see how docs look like).