Efficient classical simulation of matchgate circuits with generalized inputs and measurements

Matchgates are a restricted set of two-qubit gates known to be classically simulable under particular conditions. Specifically, if a circuit consists only of nearest-neighbour matchgates, an efficient classical simulation is possible if either (i) the input is a computational basis state and the simulation requires computing probabilities of multi-qubit outcomes (including also adaptive measurements), or (ii) if the input is an arbitrary product state, but the output of the circuit consists of a single qubit. In this paper we extend these results to show that matchgates are classically simulable even in the most general combination of these settings, namely, if the inputs are arbitrary product states, if the measurements are over arbitrarily many output qubits, and if adaptive measurements are allowed. This remains true even for arbitrary single-qubit measurements, albeit only in a weaker notion of classical simulation. These results make for an interesting contrast with other restricted models of computation, such as Clifford circuits or (bosonic) linear optics, where the complexity of simulation varies greatly under similar modifications.