Matchgate circuits produce volume-law operator entanglement and stabilizer entropy despite classical simulability

White develops a Majorana-basis Weingarten calculus for matchgate ensembles, reducing moment calculations to boundary-conditioned combinatorics. This yields exact averages for local-operator entanglement, stabilizer entropy, and higher-order OTOCs without sampling. The work demonstrates that these measures saturate to extensive values even though the circuits remain efficiently simulable via free-fermion methods.

The separation arises because operator resource measures depend on the chosen representation; Gaussian circuits can still generate high entanglement when operators are expanded beyond the initial Majorana basis. This supplies a classically tractable counterexample to the assumption that late-time OTOC decay or volume-law operator entanglement necessarily signals computational hardness, clarifying distinctions within quantum resource theories.

The result connects to doped matchgate circuits, where operator entanglement beyond the Gaussian baseline lower-bounds non-Gaussian gate count. It therefore offers a practical diagnostic for quantifying non-Gaussian resources without requiring full simulation.

Future experiments could test whether these exact matchgate averages predict resource thresholds in hardware implementations of near-Clifford or matchgate-doped circuits at intermediate system sizes.