Update the code for decomposing MCX and MCPhase gates, so that the number of CX gates won't grow exponentially

[ShellyGarion]

Summary

close #11823
see the discussion in #5872
Update the code for decomposing MCX and MCPhase gates, so that the number of CX gates won't grow exponentially

Details and comments

The current code for synthesizing MCX and MCPhase gates (without ancillas) is based on Gray decomposition and hence the number of CX gates grows exponentially in the number of qubits n.
We already have a code for MCRZ synthesis in linear number of CXs (~16*n) based on https://arxiv.org/abs/2302.06377.
In this PR, we add code for MCPhase synthesis in quadratic number of CXs (~8*n^2), and then to extend it to MCX.

Here is a comparison of the CX count in the current and new Qiskit code:

num_qubits= 4 cx count in current code= 20 cx count in new code= 20
num_qubits= 5 cx count in current code= 44 cx count in new code= 44
num_qubits= 6 cx count in current code= 92 cx count in new code= 84
num_qubits= 7 cx count in current code= 188 cx count in new code= 140
num_qubits= 8 cx count in current code= 380 cx count in new code= 220
num_qubits= 9 cx count in current code= 764 cx count in new code= 324
num_qubits= 10 cx count in current code= 1532 cx count in new code= 452
num_qubits= 11 cx count in current code= 3068 cx count in new code= 604
num_qubits= 12 cx count in current code= 6140 cx count in new code= 780
num_qubits= 13 cx count in current code= 12284 cx count in new code= 980
num_qubits= 14 cx count in current code= 24572 cx count in new code= 1204
num_qubits= 15 cx count in current code= 49148 cx count in new code= 1452
num_qubits= 16 cx count in current code= 98300 cx count in new code= 1724
num_qubits= 17 cx count in current code= 196604 cx count in new code= 2020
num_qubits= 18 cx count in current code= 393212 cx count in new code= 2340
num_qubits= 19 cx count in current code= 786428 cx count in new code= 2684
num_qubits= 20 cx count in current code= 1572860 cx count in new code= 3052

[qiskit-bot]

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

• @Cryoris
• @Qiskit/terra-core
• @ajavadia

[coveralls]

Pull Request Test Coverage Report for Build 8497005635

Warning: This coverage report may be inaccurate.

This pull request's base commit is no longer the HEAD commit of its target branch. This means it includes changes from outside the original pull request, including, potentially, unrelated coverage changes.

• For more information on this, see Tracking coverage changes with pull request builds.
• To avoid this issue with future PRs, see these Recommended CI Configurations.
• For a quick fix, rebase this PR at GitHub. Your next report should be accurate.

Details

• 28 of 28 (100.0%) changed or added relevant lines in 2 files are covered.
• 211 unchanged lines in 8 files lost coverage.
• Overall coverage decreased (-0.1%) to 89.317%

---

Files with Coverage Reduction	New Missed Lines	%
crates/qasm2/src/expr.rs	1	94.03%
qiskit/transpiler/passes/synthesis/unitary_synthesis.py	4	90.04%
crates/accelerate/src/convert_2q_block_matrix.rs	4	90.7%
crates/qasm2/src/lex.rs	8	91.6%
crates/accelerate/src/euler_one_qubit_decomposer.rs	10	89.78%
qiskit/synthesis/two_qubit/two_qubit_decompose.py	15	93.61%
crates/qasm2/src/parse.rs	18	96.69%
crates/accelerate/src/two_qubit_decompose.rs	151	89.33%

Totals
Change from base Build 8456962099:	-0.1%
Covered Lines:	60239
Relevant Lines:	67444

---

💛 - Coveralls

[ShellyGarion]

This code handles the two cases of lam differently:

• If lam is a float (or an int) then we use the mcrz synthesis to reduce the number of CX gates.

• If lam is a ParameterExpression we use the existing code since mcrz gate does not handle this type in general.

[ShellyGarion]

After completing this PR, we will be able to write a code to efficiently synthesize any multi-controlled 1-qubit unitary using Euler's decomposition in quadratic number of CX gates, based on the synthesis of mcp, mcry, and mcrz gates (this will be done in a separate PR).

[alexanderivrii]

Thanks @ShellyGarion, this leads to a significant reduction in the number of CX-gates when decomposing MCX and MCP gates. This looks good to me, with a minor nitpick about the wording in release notes. I am also wondering if this should be labeled as "new feature" rather than "bug fix".

Another question: it seems that the old method actually produced fewer CX-gates for small values of num_qubits (e.g., 5 gates instead of 6 with 2 controls). I think it would make sense to use the new method only when it's better.

[alexanderivrii]

Maybe it's slightly better to say "replace the decomposition" instead of "fix the decomposition" since the previous decomposition was technically correct. Maybe you also want to explicitly mention that this is the decomposition without ancilla qubits.

[ShellyGarion]

Another question: it seems that the old method actually produced fewer CX-gates for small values of num_qubits (e.g., 5 gates instead of 6 with 2 controls). I think it would make sense to use the new method only when it's better.

Thanks @alexanderivrii ! that's a good catch, it's due to this:

qiskit/qiskit/circuit/library/standard_gates/x.py

Line 1098 in f81e5a8

# the C3XGate and C4XGate will be implemented in the MCXGrayCode class.

Since we don't rely on the MCXGrayCode class anymore, I'll add the cases of C3XGate and C4XGate directly to the MCXGate definition.

[alexanderivrii]

Thanks Shelly for doing the updates. Having qiskit transpiler produce efficient circuits for MCX gates will make many people happy.