Summary

Fixes a problem described here: running the SolovayKitaevSynthesis unitary synthesis plugin repeatedly incorrectly reuses the set of basis gates specified on the very first run. For example,

circuit = QuantumCircuit(1)
circuit.rx(0.8, 0)
unitary = Operator(circuit).data
plugin = SolovayKitaevSynthesis()

# First, use the ["h", "t", "tdg"] set
out = plugin.run(unitary, basis_gates=["h", "t", "tdg"])
print(out.count_ops())
self.assertLessEqual(set(out.count_ops().keys()), {"h", "t", "tdg"})

# Second, use the ["h", "s"] set
out = plugin.run(unitary, basis_gates=["h", "s", "sdg"])
self.assertLessEqual(set(out.count_ops().keys()), {"h", "s", "sdg"})
print(out.count_ops())

synthesizes both times to the first specified basis set of ["h", "t", "tdg"]. Creating two different SolovayKitaevSynthesis plugins does not solve this, since the caching is done via a class-level attribute.

The fix (many thanks to @eliarbel for the suggestion!) consists of storing a dictionary from the basis gate sets to the correspoding basis approximations. UPDATE: after review and online discussion, we have decided not to store the dictionary but instead to regenerate the basic approximations whenever the set of basis gates or the depth used for generation changes from the previous run.

Needed for #14225.