Exact Operator Complexity Measures from Random Matchgate Unitaries

Abstract

Matchgate circuits describe classically tractable dynamics in a physically relevant setting while nonetheless exhibiting features characteristic of complex quantum systems, such as high amounts of magic. This invites the broader question: to what extent are typical complexity measures saturated under trivial dynamics? Here, we address this by studying the behaviour of different Heisenberg-picture complexity measures in random matchgate ensembles. Our main technical contribution is a tractable and generalisible free-fermionic unitary Weingarten calculus formulated entirely in the Majorana basis. This method is tailored to computing moments of operators, and simplifies to a combinatorics problem which depends on the boundary conditions prescribed by a given operator functional. Using this framework, we derive closed-form expressions for the typical local-operator entanglement, operator stabiliser entropy, and higher-order out-of-time-ordered correlators. For extensive initial Majorana strings, both operator entanglement and stabiliser entropy obey volume laws, even though the underlying circuits remain classically simulable. Out-of-time-ordered correlators can likewise become exponentially small, showing that their late-time saturation value alone does not witness computational hardness and constituting a classically tractable example of emergent free independence. Finally, we relate these separations to operator resource theories, showing how operator entanglement beyond the Gaussian baseline lower-bounds the number of non-Gaussian gates in doped matchgate circuits. Our results present a practical toolbox for computing exact averages over random matchgates, while clarifying the distinction between quantum resources versus simulation complexity, and their dependence on the representation being exploited.