From #9475:

@ShellyGarion 's comment is hinting that another thing that would be nice to have is something like Clifford.from_operation or similar, that could rely on methods that might be quicker than generating and checking the unitary matrix representation.

For example, checking if the name is in the list of hard coded names in _BASIS_1Q/_BASIS_2Q to know if an Instruction is Clifford, or utilizing the equivalence library to see if a Gate has an all Clifford decomposition, or for cases like the parameterized rotations where there may be a simple arithmetic check on the params to know if it is Clifford.

Additionally, the same methods could be available to know quickly if an unknown operation is_clifford.

Opening this issue to discuss how and where we can include such checks for Clifford, and for the other Operation subclasses more generally.