  Magic states, pivotal for universal quantum computation via classically
simulable Clifford gates, often undergo decomposition into resourceless
stabilizer states, facilitating simulation through classical means. This
approach yields three operationally significant metrics: stabilizer rank,
fidelity, and extent. We extend these simulation methods to encompass matchgate
circuits (MGCs), and define equivalent metrics for this setting. We begin with
an investigation into the algebraic constraints defining Gaussian states,
marking the first explicit characterisation of these states. The explicit
description of Gaussian states is pivotal to our methods for tackling all the
simulation tasks. Central to our inquiry is the concept of Gaussian rank -- a
pivotal metric defining the minimum terms required for decomposing a quantum
state into Gaussian constituents. This metric holds paramount significance in
determining the runtime of rank-based simulations for MGCs featuring magic
state inputs. The absence of low-rank decompositions presents a computational
hurdle, thereby prompting a deeper examination of fermionic magic states. We
find that the Gaussian rank of 2 instances of our canonical magic state is 4
under symmetry-restricted decompositions. Additionally, our numerical analysis
suggests the absence of low-rank decompositions for 2 or 3 copies of this magic
state. Further, we explore the Gaussian extent, a convex metric offering an
upper bound on the rank. We prove the Gaussian extent's multiplicative
behaviour on 4-qubit systems, along with initial strides towards proving its
sub-multiplicative nature in general settings. One important result in that
direction we present is an upper bound on the Gaussian fidelity of generic
states.
