Matchgate simulator

[kevinsung]

What should we add?

A circuit consisting only of Clifford gates can be efficiently simulated using the StabilizerState class. There is another important class of classically simulable circuits, which are those consisting of matchgates acting between nearest-neighbor qubits on a line (see https://arxiv.org/abs/0804.4050). In the context of Qiskit, matchgates are represented by XXPlusYYGate and XXMinusYYGate. My request is to add functionality to efficiently simulate these circuits.

[ShellyGarion]

Generally, we don't have simulators in Qiskit (except of the basic ones: Statevector and StabilizerState) only in qiskit-aer.
Could you perhaps elaborate on the importance of this simulator?

[trigpolynom]

I'd like to help with this if possible. I am interested in classical simulation in general and would like to learn more about matchgates.