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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2603.12392 |
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| _version_ | 1866911511358734336 |
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| author | Sierant, Piotr Turkeshi, Xhek Tarabunga, Poetri Sonya |
| author_facet | Sierant, Piotr Turkeshi, Xhek Tarabunga, Poetri Sonya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12392 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Theory of the Matchgate Commutant Sierant, Piotr Turkeshi, Xhek Tarabunga, Poetri Sonya Quantum Physics Statistical Mechanics Mathematical Physics 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. |
| title | Theory of the Matchgate Commutant |
| topic | Quantum Physics Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2603.12392 |