Classical simulation of dissipative fermionic linear optics

Fermionic linear optics is a limited form of quantum computation which is known to be efficiently simulable on a classical computer. We revisit and extend this result by enlarging the set of available computational gates: in addition to unitaries and measurements, we allow dissipative evolution governed by a Markovian master equation with linear Lindblad operators. We show that this more general form of fermionic computation is also simulable efficiently by classical means. Given a system of $N$ fermionic modes, our algorithm simulates any such gate in time $O(N^3)$ while a single-mode measurement is simulated in time $O(N^2)$. The steady state of the Lindblad equation can be computed in time $O(N^3)$.