  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 $O(N^3)$.
