A Lightweight Protocol for Matchgate Fidelity Estimation

We present a low-depth randomised protocol for estimating the entanglement fidelity between an $n$-qubit matchgate circuit $\mathcal{U}$ and its noisy implementation $\mathcal{E}$. Our method uses a Pauli-Liouville representation adapted to Clifford algebra elements, in which matchgate superoperators acquire a block-diagonal form. This structure enables efficient direct fidelity estimation using only Pauli state preparation and measurement, while avoiding the exponentially costly sampling step required for generic unitary channels. Compared with the protocol of Flammia and Liu [PRL 106, 230501], our algorithm gives an exponential improvement in classical sampling complexity and a multiplicative $1/\sqrt{n}$ reduction in expected shot count for matchgate circuits. The protocol also extends, without asymptotic overhead, to matchgate circuits sandwiched between Clifford circuits. For nearest-neighbour $XY(\theta)$ gates and Givens rotations we demonstrate an increase in superoperator sparsity, giving an additional $1/\sqrt{n}$ reduction in expected shot count and, to our knowledge, the first scalable fidelity estimation protocol for these important matchgate subgroups.