Toward a quasi-probability representation of matchgate circuits

Quantum circuits composed of a particular class of gates called matchgates range from circuits that are classically simulatable to those that can perform universal quantum computation. Matchgate computation can also be understood from a more physical point of view as a computation with fermionic modes. We attempt to construct a quasi-probability (phase space) representation of quantum theory in which classically simulatable matchgate circuits are represented positively i.e. non-contextually.