Classical simulation of fermionic linear optics augmented with noisy ancillas

Fermionic linear optics is a model of quantum computation which is efficiently simulable on a classical probabilistic computer. We study the problem of a classical simulation of fermionic linear optics augmented with noisy auxiliary states. If the auxiliary state can be expressed as a convex combination of pure fermionic Gaussian states, the corresponding computation scheme is classically simulable. We present an analytic characterization of the set of convex-Gaussian states in the first nontrivial case, in which the Hilbert space of the ancilla is a four-mode Fock space. We use our result to solve an open problem recently posed by de Melo et al. [New J. Phys. 15, 013015 (2013)] and to study in detail the geometrical properties of the set of convex-Gaussian states.