Extending Matchgate Simulation Methods to Universal Quantum Circuits

Matchgates are a family of parity-preserving two-qubit gates, nearest-neighbour circuits of which are known to be classically simulable in polynomial time. In this work, we present a simulation method to classically simulate an $\boldsymbol{n}$-qubit circuit containing $\boldsymbol{N}$ matchgates and $\boldsymbol{m}$ universality-enabling gates in the setting of single-qubit Pauli measurements and product state inputs. The universality-enabling gates we consider include the SWAP, CZ and CPhase gates. We find in the worst and average cases, the scaling when $\boldsymbol{m \, < \, \lfloor \frac{n}{2} \rfloor -1}$ is given by $\sim \mathcal{O}(\boldsymbol{N(\frac{n}{m+1})^{2m+2}})$ and $\sim \mathcal{O}( \boldsymbol{\frac{N}{m+1}(\frac{n}{m+1})^{2m+2}})$, respectively. For $\boldsymbol{m \, \geq \, \lfloor \frac{n}{2} \rfloor -1}$, we find the scaling is exponential in $\boldsymbol{n}$, but always outperforms a dense statevector simulator in the asymptotic limit.