Automatically set and save optimal OpenMP and fusion thresholds

[hhorii]

Summary

Adds basic profiling code to find suitable threshold values for OpenMP and gate fusion for backend options. This is a revised version of #956.

Details and comments

This PR will add following into to #956.

• [x] Save optimized parameters into settings.conf
• [x] Optimize fusion_cost.N

[hhorii]

Once https://github.com/Qiskit/qiskit-terra/pull/5943 is merged, optimized parameters will be able to be store in user config around here: https://github.com/Qiskit/qiskit-aer/blob/9f7b832abf8c74ed6d04d62072ff000deae882ab/qiskit/providers/aer/profile.py#L68-L69 https://github.com/Qiskit/qiskit-aer/blob/9f7b832abf8c74ed6d04d62072ff000deae882ab/qiskit/providers/aer/profile.py#L82-L84

[hhorii]

Here are performance graphs (time_taken includes only transpilation + simulation and excludes interface overheads) with default and tuned configurations.

Configurations:

• Left: MacBook (2.6 GHz 6-Core Intel Core i7)
  • statevector_parallel_threshold=11
  • fusion_threshold=19
• Right: Xeon (Intel Xeon Gold 6140 CPU )
  • statevector_parallel_threshold=16
  • fusion_threshold=11

QuantumVolume:

IntegerComparator:

WeightedAdder:

QuadraticForm:

QFT:

RealAmplitudes with full entanglements

RealAmplitudes with linear entanglements

Red lines are expected the lowest. Because this profiling function uses QuantumVolume, tuned options do not work for others. However, I believe, these side-effects are acceptables.