Qiskit-Extensions / qiskit-experiments / 6099303896

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Improve performance of RB circuit generation (#1263)

### Summary

The circuit generation for RB experiments is slow due to the use of
indexing on sparse matrices (`_CLIFFORD_COMPOSE_2Q` from
`qiskit_experiments.library.randomized_benchmarking.clifford_utils`).
Since indexing is the only operation used on `_CLIFFORD_COMPOSE_2Q` we
can improve performance by using the `lil_matrix` format.

### Details and comments

Improve performance of RB circuit generation by changing the format of
`_CLIFFORD_COMPOSE_2Q` to `lil_matrix`.

Micro benchmark:
```
import scipy.sparse
from qiskit_experiments.library.randomized_benchmarking.clifford_utils import _CLIFFORD_COMPOSE_2Q

_CLIFFORD_COMPOSE_2Q_lil =  scipy.sparse.lil_matrix(_CLIFFORD_COMPOSE_2Q)

%timeit _CLIFFORD_COMPOSE_2Q[2,3]
%timeit _CLIFFORD_COMPOSE_2Q_lil[2,3]
```
Result:
```
29.4 µs ± 4.01 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
2.64 µs ± 777 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
```

Application benchmark:
```
import time
import qiskit
from qiskit_experiments.library import InterleavedRB

gate=qiskit.circuit.library.CXGate()
rb=InterleavedRB(gate, physical_qubits= (0,1),lengths= [1,10,50,100,200], num_samples= 10)

t0=time.time()
c=rb.circuits()
dt=time.time()-t0
```
results in a speedup from 2.1419544219970703 to 1.154689073562622
seconds.



### PR checklist (delete when all criteria are met)

- [x] I have read the contributing guide `CONTRIBUTING.md`.
- [x] I have added the tests to cover my changes.
- [x] I have updated the documentation accordingly.
- [x] I have added a release note file using `reno` if this change needs
to be documented in the release notes. (no release note required)

---------

Co-authored-by: Helena Zhang <Helena.Zhang@ibm.com>

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