Theory of the Matchgate Commutant

• URL: http://arxiv.org/abs/2603.12392v1
• Date: Thu, 12 Mar 2026 19:12:19 GMT
• Title: Theory of the Matchgate Commutant
• Abstract summary: We show that operators coupling different system copies generate the Lie algebra $mathfrakso(k)$, endowing the space of invariants with rich and tractable structure.<n>We also characterize the commutant of the Clifford-matchgate subgroup, showing that restricting to signed permutations of Majorana modes yields a commutant that qualitatively diverges from the matchgate case for $k geq 4$ replicas.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In quantum information theory and statistical physics, symmetries of multiple copies, or replicas, of a system play a pivotal role. For unitary ensembles, these symmetries are encoded in the replicated commutant: the algebra of operators commuting with the ensemble across $k$ replicas. Determining the commutant is straightforward for the full unitary group, but remains a major obstacle for structured, computationally relevant circuit families. We solve this problem for matchgate circuits, which prepare fermionic Gaussian states on $n$ qubits. Using a Majorana fermion representation, we show that operators coupling different system copies generate the orthogonal Lie algebra $\mathfrak{so}(k)$, endowing the space of invariants with rich and tractable structure. This underlying symmetry decomposes the matchgate commutant into irreducible sectors, which we completely resolve via a Gelfand--Tsetlin construction. We provide an explicit orthonormal basis of the matchgate commutant for all $k$ and $n$, together with a formula for its dimension that grows polynomially in $n$. Furthermore, we characterize the commutant of the Clifford--matchgate subgroup, showing that restricting to signed permutations of Majorana modes yields a commutant that qualitatively diverges from the matchgate case for $k \geq 4$ replicas. Ultimately, our orthonormal basis turns algebraic classification into a working toolbox. Using it, we derive closed-form expressions for matchgate twirling channels and a fermionic analogue of Weingarten calculus, the projector encoding all moments of the Gaussian state orbit, state and unitary frame potentials, the average nonstabilizerness of fermionic Gaussian states, a systematic hierarchy of non-Gaussianity measures, and a fermionic de Finetti theorem.

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