  Fermionic linear optics is efficiently classically simulatable. Here it is
shown that the set of states achievable with fermionic linear optics and
particle measurements is the closure of a low dimensional Lie group. The
weakness of fermionic linear optics and measurements can therefore be explained
and contrasted with the strength of bosonic linear optics with particle
measurements. An analysis of fermionic linear optics is used to show that the
two-qubit matchgates and the simulatable matchcircuits introduced by Valiant
generate a monoid of extended fermionic linear optics operators. A useful
interpretation of efficient classical simulations such as this one is as a
simulation of a model of non-deterministic quantum computation. Problem areas
for future investigations are suggested.
