Adding permutation synthesis algorithm for LNN

[alexanderivrii]

Summary

This commit add the depth-efficient synthesis algorithm for Permutation over the linear nearest-neighbor architecture (LNN) following the paper https://arxiv.org/abs/quant-ph/0701194, chapter 6. This is the second task in #9036.

Any permutation can be synthesized over LNN using a network of SWAP gates with depth at most n and size at most n(n-1)/2 (where both depth and size are measured with respect to SWAPs), or equivalently with depth at most 3n and size at most 3n(n-1)/2 (measured with respect to CNOTs).

Additional (minor) changes include improving the documentation of what a "permutation pattern" is, and moving _get_ordered_swaps to qiskit/synthesis/permutation/permutation_utils.py (and test_get_ordered_swap to test/python/circuit/library/test_permutation.py.

The next step (that would be handled in a follow-up PR) is to add a high-level synthesis plugin for Permutation (this may also require changing Permutation from QuantumCircuit to Gate).

Co-authored with https://github.com/NirGavrielov (Nir Gavrielov).

[coveralls]

Pull Request Test Coverage Report for Build 3490868680

• 37 of 37 (100.0%) changed or added relevant lines in 5 files are covered.
• 2 unchanged lines in 1 file lost coverage.
• Overall coverage increased (+0.04%) to 84.614%

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Files with Coverage Reduction	New Missed Lines	%
src/sabre_swap/layer.rs	2	98.95%

Totals
Change from base Build 3490867390:	0.04%
Covered Lines:	62690
Relevant Lines:	74089

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💛 - Coveralls

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[ShellyGarion]

LGTM, thanks!

[mtreinish]

Oh, also I realized I think we're not adding this new function to the doc tree anywhere we probably should add it so we're publishing the documentation for this algorithm.

[alexanderivrii]

@mtreinish, thanks for the comments, it took me quite a number of attempts to get the docstrings working, but now all seems to be good.

[mtreinish]

Thanks for making the updates, in my previous review. Looking through the code again I caught one potential inefficiency which should be easy to fix. Other than that I think this probably good to go.

[mtreinish]

I would avoid using the index() call here, it basically is an O(n) lookup on each iteration making this loop scale quadratically. If you need to get the index of i from permutation, I would build a map outside the loop that can do this look up in constant time. That way this function is O(2n) instead of O(n**2).

[alexanderivrii]

This is a good point. For the record, this function existed before, I have moved it to permutation_utils.py without any modifications. But I agree that it would be nice to make it more efficient. Hmm, it might be just a tiny bit trickier than what you have suggested, as the permutation (and hence also index()) changes throughout the loop, but it's clear that a linear algorithm should be possible.

[alexanderivrii]

And that was very easy, done in 15f1228.

[mtreinish]

LGTM, thanks for the quick update.