Unitary Designs from Doped Matchgate Circuits

• URL: http://arxiv.org/abs/2606.23800v1
• Date: Mon, 22 Jun 2026 18:00:04 GMT
• Title: Unitary Designs from Doped Matchgate Circuits
• Abstract summary: We study a controlled route by doping matchgate circuits with non-Gaussian gates-physically.<n>We find approximate parity-preserving $2$-designs in polylogarithmic depth with a sparse non-Gaussian gate count.<n>Our results establish doped matchgate circuits as a controlled, analytically tractable route from free fermions to interaction-generated quantum designs.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Matchgate circuits realize free-fermion dynamics: they are efficiently classically simulable, yet cannot on their own generate the generic randomness required for universal computation or unitary design formation. We study a controlled route beyond this integrable limit by doping matchgate circuits with non-Gaussian gates-physically, the injection of fermionic interactions into an otherwise free system. Using the matchgate commutant framework, we obtain analytic control over unitary $2$-design formation. For globally scrambled dynamics, the design problem maps exactly onto a classical birth-death Markov chain with an Ornstein-Uhlenbeck continuum limit, recasting the emergence of quantum randomness in terms of spectral gaps and mixing times and yielding rigorous bounds on the number of non-Gaussian gates needed for approximate $2$-designs. These bounds hold for a broad class of parity-preserving non-Gaussian gates, independently of microscopic details, with numerics indicating that the same mechanism governs higher-order designs. Used as local building blocks in a glued-circuit architecture, they yield approximate parity-preserving $2$-designs in polylogarithmic depth with a sparse non-Gaussian gate count, with implications for Page-like entanglement growth and fermionic classical-shadow protocols. Finally, locality reshapes this picture: in local brickwork dynamics, design formation is diffusion-limited and far slower. Our results establish doped matchgate circuits as a controlled, analytically tractable route from free fermions to interaction-generated quantum designs.

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