Decomposition of single R gate in RR basis should be one R gate, not two

[sven-oxionics]

Summary

Currently, a single R gate may come out of RR-decomposition as two R gates, which is obviously not ideal.

This PR adds test cases for when this happens and contains the code changes to only emit one R gate when possible.

Details and comments

For example, RGate(0.1, 0.2) is currently decomposed (in RR basis) as

   ┌────────────────┐┌──────────┐
0: ┤ R(-3.0416,0.2) ├┤ R(π,0.2) ├
   └────────────────┘└──────────┘

Two R(𝜗, 𝜑) gates with the same 𝜑 parameter can be combined into one by simply adding up the 𝜗 values (giving us R(0.1, 0.2), unsurprisingly).

In terms of U(𝜗, 𝜑, 𝜆), it is the case when 𝜑 = -𝜆 that the two R-gates we construct have the same second parameter and therefore should be expressed as a single R gate. For example, U3Gate(0.1, 0.2, -0.2) currently produces this RR-decomposition:

   ┌───────────────────┐┌─────────────┐
0: ┤ R(-3.0416,1.7708) ├┤ R(π,1.7708) ├
   └───────────────────┘└─────────────┘

which also unnecessarily consists of two R gates instead of just one.

This commit adds the two examples above as unit tests, ensuring they RR-decompose two just one R gate, as well as the code changes to make these two new tests pass, along with all existing tests, of course.

The condition for this special case is that the 𝜑 parameters of the two R-gates we would emit are the same (thus mod_2pi(PI / 2. - lam)=mod_2pi(0.5 * (phi - lam + PI), simplified as mod_2pi((phi + lam) / 2)=0).

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[coveralls]

Pull Request Test Coverage Report for Build 7030458574

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A quick fix for this PR: rebase it; your next report should be accurate.

• 11 of 11 (100.0%) changed or added relevant lines in 1 file are covered.
• 158 unchanged lines in 34 files lost coverage.
• Overall coverage increased (+0.8%) to 87.242%

---

Files with Coverage Reduction	New Missed Lines	%
crates/qasm2/src/lex.rs	1	92.17%
qiskit/circuit/library/generalized_gates/pauli.py	1	91.18%
qiskit/circuit/parametertable.py	1	87.4%
qiskit/providers/backend_compat.py	1	86.05%
qiskit/providers/backend.py	1	81.08%
qiskit/providers/basicaer/basicaertools.py	1	91.23%
qiskit/providers/basicaer/qasm_simulator.py	1	85.42%
qiskit/providers/job.py	1	56.9%
qiskit/quantum_info/states/statevector.py	1	89.69%
qiskit/synthesis/evolution/product_formula.py	1	97.56%

Totals
Change from base Build 6966472058:	0.8%
Covered Lines:	60785
Relevant Lines:	69674

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💛 - Coveralls

[jakelishman]

Thanks for this! Sorry for the slightly delayed response. It's great to get these inefficiencies ironed out in the less-used decomposition targets.

Would you mind adding a feature release note mentioning the improved synthesis quality (reno new --edit rr-decomposition-synthesis or similar)? I can do it if it's easier, though.

[sven-oxionics]

No problem, @jakelishman! I had a go. Feel free to edit if it doesn't quite sit right.

[jakelishman]

This looks fine to me, thanks for finding it, and for contributing the fix!