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Paolo Braccia, N. L. Diaz, Martin Larocca, M. Cerezo, Diego García-Martín (Jun 02 2025).
Abstract: Sampling unitary Fermionic Linear Optics (FLO), or matchgate circuits, has become a fundamental tool in quantum information. Such capability enables a large number of applications ranging from randomized benchmarking of continuous gate sets, to fermionic classical shadows. In this work, we introduce optimal algorithms to sample over the non-particle-preserving (active) and particle-preserving (passive) FLO Haar measures. In particular, we provide appropriate distributions for the gates of nn-qubit parametrized circuits which produce random active and passive FLO. In contrast to previous approaches, which either incur classical O(n3)\mathcal{O}(n^3) compilation costs or have suboptimal depths, our methods directly output circuits which simultaneously achieve an optimal down-to-the-constant-factor Θ(n)\Theta(n) depth and Θ(n2)\Theta(n^2) gate count; with only a Θ(n2)\Theta(n^2) classical overhead. Finally, we also provide quantum circuits to sample Clifford FLO with an optimal Θ(n2)\Theta(n^2) gate count.

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