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
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≥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.
Understanding the structure of operators that commute with
k identical replicas of unitary ensembles, also known as their
k-commutants, is an important problem in quantum many-body physics with deep implications for the late-time behavior of physical quantities such as correlation functions and entanglement entropies under unitary evolution. In this work, we study the
k-commutants of free-fermion unitary systems, which are heuristically known to contain
SO(k) and
SU(k) groups without and with particle number conservation respectively, with formal derivations of projectors onto these commutants appearing only very recently. We establish a complementary perspective by highlighting a larger
O(2k) replica symmetry (or
SU(2k) respectively) that the
k-commutant transforms irreducibly under, which leads to a simple geometric understanding of the commutant in terms of coherent states parametrized by a Grassmannian manifold. We derive this structure by mapping the
k-commutant to the ground state of effective ferromagnetic Heisenberg models, analogous to the ones that appear in the noisy circuit literature, which we solve exactly using standard representation theory methods. Further, we show that the Grassmannian manifold of the
k-commutant is exactly the manifold of fermionic Gaussian states on
2k sites, which reveals a duality between real space and replica space in free-fermion systems. This geometric understanding also provides a compact projection formula onto the
k-commutant, based on the resolution of identity for coherent states, which can prove advantageous in analytical calculations of averaged non-linear functionals of Gaussian states, as we demonstrate using some examples for the entanglement entropies. In all, this work provides a geometric perspective on the
k-commutant of free-fermions that naturally connects to problems in quantum many-body physics.
We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. For the Haar ensemble of Majorana Gaussian states we derive closed finite-
N expressions for the average stabilizer purities and reconstruct the Pauli spectrum exactly, as a mixture of products of beta-distributed variables resolved by Majorana weight. The stabilizer entropies, and their filtered versions, freeze at Rényi index
qc=2: for
q>2 the magic density is
1/(q−1), controlled by rare weight-two Majorana strings. Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer
q as coefficient integrals over
SU(2q), with dimension independent of
N, and evaluate the
q=2 case exactly. We establish that the frozen density describes typical states in the Majorana and half-filled number-conserving ensembles and relate these ensembles to the corresponding
SYK2 ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction
κ<κ⋆G≃0.7613 of their modes in the thermodynamic limit, remaining magical even beyond the
2/3 threshold for Haar-random states in the full Hilbert space.
Fermionic Gaussian states describe free fermions and underlie the mean-field picture of matter, from metals to superconductors; they are also efficiently simulable on classical computers. Departures from Gaussianity -- the correlations produced by interactions -- are therefore what make a fermionic system hard to simulate classically and useful for quantum computation, analogous to the role of magic in stabilizer-based quantum computation. Yet detecting and quantifying such non-Gaussianity at scale has remained challenging. Here we introduce practical tests and witnesses of fermionic non-Gaussianity built on fermionic antiflatness, a measure derived from the two-point covariance matrix. We estimate it with two protocols -- a two-copy Bell measurement and a single-copy scheme using commuting Majorana bilinears -- that determine whether a state is Gaussian or far from it at lower measurement cost than existing approaches, using only operations native to fault-tolerant hardware. For mixed states, a purity-corrected witness certifies non-Gaussianity and remains robust under strong noise; running it on the IQM quantum processor, we find that noise can both reduce and enhance non-Gaussianity. Finally, we show that preparing pseudorandom fermionic states requires extensive non-Gaussianity. Together, these tools enable the study and certification of non-Gaussian fermionic resources on present-day quantum devices.
Fermionic Gaussian states form a central class of classically tractable quantum states, while fermionic non-Gaussianity provides the resource required to go beyond free-fermion dynamics. A key challenge is to quantify this resource through monotones that are both mathematically rigorous and experimentally accessible. Here, we show that the fermionic entropy, defined through the squared Frobenius norm of the correlation matrix, is a strong pure-state Gaussian monotone. Its simple closed-form expression also makes it directly measurable: we show that the associated fermionic purity can be unbiasedly estimated up to additive error
ε using
O(ε−2) two-copy measurements, independently of the system size. Moreover, we prove that the fermionic entropy obeys asymptotic continuity and, as a direct consequence, establish its operational meaning as the upper bound to the asymptotic rate of non-Gaussianity distillation. We further derive a linear sample complexity bound for tolerant testing of fermionic Gaussian states, providing a quadratic improvement over the state of the art. As a further application of our results, we study unitary designs generated by Matchgate circuits supplemented with Majorana-local non-Gaussian gates. We prove that a linear number of such gates is necessary even to achieve an approximate state
2-design with error below
0.4%. Combined with known nearly linear upper bounds for relative-error designs, this determines the optimal doping level, up to logarithmic factors, across all relevant design notions and reveals the extensive non-Gaussianity cost required to generate Haar-like quantum dynamics in this architecture.
Fermionic Gaussian operators are foundational tools in quantum many-body theory, numerical simulation of fermionic dynamics, and fermionic linear optics. While their structure is fully determined by two-point correlations, evaluating their matrix elements in arbitrary local spin bases remains a nontrivial task, especially in applications involving quantum measurements, tomography, and basis-rotated simulations. In this work, we derive a fully explicit and general Pfaffian formula for the matrix elements of fermionic Gaussian operators between arbitrary Pauli product states. Our approach introduces a pair of sign-encoding matrices whose classification leads to a Lie algebra isomorphic to
so(2L). This algebraic structure not only guarantees consistency of the Pfaffian signs but also reveals deep connections to Clifford algebras. The resulting framework enables scalable computations across diverse fields -- from quantum tomography and entanglement dynamics to algebraic structure in fermionic circuits and matchgate computation. Beyond its practical utility, our construction sheds light on the internal symmetries of Gaussian operators and offers a new lens through which to explore their role in quantum information and computational models.