QFT circuit synthesis for linear nearest neighbor connectivity

[ShellyGarion]

Summary

#9036

Synthesis of QFT circuits for linear nearest neighbor connectivity
based on Fowler et al, Fig 2.b from https://arxiv.org/abs/quant-ph/0402196

Details and comments

[qiskit-bot]

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

• @Qiskit/terra-core

[coveralls]

Pull Request Test Coverage Report for Build 7083208146

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• 29 of 29 (100.0%) changed or added relevant lines in 3 files are covered.
• 727 unchanged lines in 70 files lost coverage.
• Overall coverage increased (+1.6%) to 87.454%

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Files with Coverage Reduction	New Missed Lines	%
qiskit/circuit/classical/types/types.py	1	96.43%
qiskit/circuit/controlflow/control_flow.py	1	95.0%
qiskit/circuit/controlflow/for_loop.py	1	98.55%
qiskit/circuit/controlflow/while_loop.py	1	98.18%
qiskit/circuit/library/generalized_gates/pauli.py	1	91.18%
qiskit/circuit/parameter.py	1	98.31%
qiskit/circuit/parametertable.py	1	87.4%
qiskit/converters/circuit_to_instruction.py	1	97.67%
qiskit/exceptions.py	1	92.31%
qiskit/providers/backend_compat.py	1	86.05%

Totals
Change from base Build 6911016807:	1.6%
Covered Lines:	59955
Relevant Lines:	68556

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💛 - Coveralls

[ShellyGarion]

Here are some performance benchmarks done with @alexanderivrii (on ibm_brisbane device)

10-qubit QFT circuits:

Original QFT code (do_swaps=False)
Transpiled using Qiskit: depth = 523, count-2q = 172, depth-2q = 111
Transpiler using TKet:   depth = 323, count-2q = 169, depth-2q = 91
QFT-LNN code (do_swaps = True)
Transpiled using Qiskit: depth = 252, count-2q = 135, depth-2q = 51
Transpiler using TKet:   depth = 269, count-2q = 149, depth-2q = 77

50-qubit QFT circuits:

Original QFT code (do_swaps=False)
Transpiled using Qiskit: depth = 4552, count-2q = 4940, depth-2q = 951
Transpiler using TKet:   depth = 3040, count-2q = 4267, depth-2q = 910
QFT-LNN code (do_swaps = True)
Transpiled using Qiskit: depth = 1458, count-2q = 3675, depth-2q = 291
Transpiler using TKet:   depth = 3036, count-2q = 4265, depth-2q = 910

[Cryoris]

Some minor comments, otherwise looks good!

[Cryoris]

LGTM, just left some tiny comments above. The one thing I'm still not quite understanding is how this has the approximately same number of gates as the tket example but is way shallower. But as long as the synthesis is correct I guess that's good 🙂