Synthesis of a Clifford circuit into layers

[ShellyGarion]

Summary

This PR decompose a Clifford into 8 layers of 1-qubit and 2-qubit gates.
There are two layers of CZ gates, and one layer of CX gates.
This PR provides a similar construction to Lemma 8 of https://arxiv.org/abs/2003.09412.

co-authored by @alexanderivrii.

Details and comments

The aim is to decompose a Clifford for LNN (linear nearest neighbor) connectivity, see #9036

This PR will be followed by two other PRs for LNN connectivity:

• Decompose a CZ circuit in depth 2n by https://arxiv.org/abs/1705.09176

• Decompose a CX-CZ circuit in depth 5n by https://arxiv.org/abs/2210.16195

Both of these PRs will allow to decompose a Clifford circuit in depth 7n over LNN connectivity.

This code has been used in: https://arxiv.org/abs/2212.03937

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[coveralls]

Pull Request Test Coverage Report for Build 3890612257

• 150 of 169 (88.76%) changed or added relevant lines in 3 files are covered.
• 2 unchanged lines in 1 file lost coverage.
• Overall coverage increased (+0.04%) to 84.641%

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Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
qiskit/synthesis/clifford/clifford_decompose_layers.py	136	155	87.74%

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

Totals
Change from base Build 3889495451:	0.04%
Covered Lines:	64472
Relevant Lines:	76171

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💛 - Coveralls

[alexanderivrii]

Thanks Shelly for driving this development. This is an important decomposition for Clifford circuits that I know many people are eagerly waiting for being available in Qiskit.

I am not sure that I should be the one officially approving this PR, since I am listed as a co-author. I have included a few minor comments, mostly about documentation.

I am also no longer sure that we need the new LayeredCircuit class. Initially it seemed a good idea since the decomposition algorithm produces the final circuit which consists of multiple subcircuits (aka "layers"), including a layer consisting only of Hadamard gates, a layer consisting only of CX-gates. a layer consisting only of CZ-gates, etc. And thus it seemed a good idea that we should first create such a layered decomposition, and then apply layer-specific synthesis algorithms to synthesize each layer (e.g., either Patel-Markov-Hayes or Kutin-Moulton-Smithline linear function synthesis algorithm for CX-layers, a CZ-synthesis algorithm for CZ-layers, etc.), but now we are synthesizing these layers eagerly anyway. Bottom line: let's remove the LayeredCircuit class and simply keep a list of QuantumCircuits as you originally had.

Another question is whether we want to use HighLevelSynthesis to synthesize layers instead of calling specific cx_synth_func, etc.? Oh, I have realized that one does not prevent the other, e.g., we can pass cx_synth_func that simply replaces a "CX layer" by LinearFunction, and then HLS would be applicable as a follow-up stage. Which means: there is nothing extra needed to do.

[eliarbel]

Overall LGTM, with the caveat I didn't get to check the algorithm details... But then I count the unit test to suffice for this purpose. I left a couple minor comments but I don't think those are blockers for merge

[alexanderivrii]

Thanks Shelly! Since I am listed as a co-author on this PR (as I have helped a bit with the initial prototype), I have asked Eli for additional comments. Given that you have also fully addressed his comments, I am marking this as approved.