  We present a classical algorithm for simulating universal quantum circuits
composed of "free" nearest-neighbour matchgates or equivalently
fermionic-linear-optical (FLO) gates, and "resourceful" non-Gaussian gates. We
achieve the promotion of the efficiently simulable FLO subtheory to universal
quantum computation by gadgetizing controlled phase gates with arbitrary phases
employing non-Gaussian resource states. Our key contribution is the development
of a novel phase-sensitive algorithm for simulating FLO circuits. This allows
us to decompose the resource states arising from gadgetization into free states
at the level of statevectors rather than density matrices. The runtime cost of
our algorithm for estimating the Born-rule probability of a given quantum
circuit scales polynomially in all circuit parameters, except for a linear
dependence on the newly introduced FLO extent, which scales exponentially with
the number of controlled-phase gates. More precisely, as a result of finding
optimal decompositions of relevant resource states, the runtime doubles for
every maximally resourceful (e.g., swap or CZ) gate added. Crucially, this cost
compares very favourably with the best known prior algorithm, where each swap
gate increases the simulation cost by a factor of approximately 9. For a
quantum circuit containing arbitrary FLO unitaries and $k$ controlled-Z gates,
we obtain an exponential improvement $O(4.5^k)$ over the prior
state-of-the-art.
