Fermionic Gaussian states are completely characterized by their two-point correlations, making the covariance matrix a natural reference for quantifying non-Gaussian structure. Motivated by fermionic antiflatness, we study its extremal form for pure number-conserving states and ask which states can be maximally antiflat and how they can be constructed. We show that, at fixed particle number, maximal first antiflatness is equivalent to a completely flat one-body reduced density matrix, and we obtain a complete existence classification together with explicit block and cyclic-orbit constructions of all allowed sectors. We then investigate the structure that appears beyond the covariance level. Maximally antiflat states with identical one-body data can exhibit distinct connected four-point correlations, which we characterize through the four-point Majorana Wick error and, equivalently, the connected two-body reduced density matrix cumulant. We show that this quartic structure is necessarily nonzero and formulate its extremization as a two-body reduced density matrix (2RDM) purity problem. Combining this framework with a Hilbert--Schmidt estimate of Christiansen for truncated two-body correlations, we obtain an upper bound on the connected quartic structure in terms of first antiflatness. Our results establish both a complete characterization of maximal fermionic antiflatness at fixed particle number and a constrained hierarchy linking covariance-level flatness to higher-order non-Gaussian correlations.
Fermionic Gaussian states are completely characterized by their two-point correlations, making the covariance matrix a natural reference for quantifying non-Gaussian structure. Motivated by fermionic antiflatness, we study its extremal form for pure number-conserving states and ask which states can be maximally antiflat and how they can be constructed. We show that, at fixed particle number, maximal first antiflatness is equivalent to a completely flat one-body reduced density matrix, and we obtain a complete existence classification together with explicit block and cyclic-orbit constructions of all allowed sectors. We then investigate the structure that appears beyond the covariance level. Maximally antiflat states with identical one-body data can exhibit distinct connected four-point correlations, which we characterize through the four-point Majorana Wick error and, equivalently, the connected two-body reduced density matrix cumulant. We show that this quartic structure is necessarily nonzero and formulate its extremization as a two-body reduced density matrix (2RDM) purity problem. Combining this framework with a Hilbert--Schmidt estimate of Christiansen for truncated two-body correlations, we obtain an upper bound on the connected quartic structure in terms of first antiflatness. Our results establish both a complete characterization of maximal fermionic antiflatness at fixed particle number and a constrained hierarchy linking covariance-level flatness to higher-order non-Gaussian correlations.
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.
Pure-state entanglement rests on a single algebraic backbone: majorization of the Schmidt spectrum governs state conversion under local operations and classical communication, and constrains entanglement monotones. Here we establish a corresponding majorization law for fermionic non-Gaussianity, the resource that elevates free fermions to universal quantum computation. Under any fermionic Gaussian protocol with pure state outcomes, the Williamson spectrum of a pure state's Majorana covariance matrix is weakly majorized by its ensemble average. This spectral law mirrors that of entanglement theory. It turns computable non-Gaussianity quantifiers such as fermionic antiflatness and occupation entropies into strong monotones for fermionic non-Gaussianity, and delivers necessary conditions and converse bounds on state conversion under Gaussian protocols. When fermion parity is conserved, no catalyst can remove a majorization obstruction---unless it carries parity coherence---and asymptotic interconversion is irreversible already for pure states. All relevant quantities are accessible from two-point Majorana correlators, turning the theory developed here into experimentally observable properties of quantum matter, testable on present-day quantum devices.
We explore the concept of antiflatness to characterize the structural fluctuations within the entanglement spectrum of a quantum state (i.e., the spectrum of its reduced density operator). As a measure of the interplay between entanglement and magic, two fundamental quantum resources, antiflatness provides second-order information about quantum correlations that standard average measures fail to capture. Recognizing that standard majorization theory fundamentally orders states by purity and is structurally blind to spectral fluctuations, we introduce a novel partial ordering known as antiflat majorization, based on the Rényi entropy spread. We define Flatness-Preserving Operations (FPOs), establishing new necessary conditions for state convertibility. Furthermore, we unify different measures of antiflatness-such as Capacity of Entanglement, Linear Rényi spread, and Logarithmic antiflatness-using the frameworks of escort distributions and Bregman divergences. We prove that the Capacity of Entanglement can be expressed as a second derivative of the Kullback-Leibler divergence along the escort trajectory, connecting it with the Quantum Fisher Information. Finally, we demonstrate that absolute maximal antiflatness is not achieved by a single universal state, but rather by a continuous Pareto frontier of extremal states with jump spectra, and we analyze the typicality of these spectral fluctuations using Haar, Bures-Hall and t-doped Clifford random state ensembles.
A recurring principle in many-body physics is that sufficiently distributed interactions or correlations lead to an effectively mean-field description. For fermionic Gaussian states, we establish a quantitative version of this principle directly at the level of Majorana covariance matrices. The main observation is that the admissibility condition for fermionic covariance matrices imposes a quantitative monogamy constraint on two-point correlations: a fixed region has only a bounded covariance budget to distribute among many disjoint regions. We use this principle in three directions. First, for quadratic fermionic Hamiltonians on finite simple graphs of high degree, we obtain product-state approximations to both the ground-state energy and the finite-temperature free energy, with error of order D−1/2 for D-regular graphs. Second, we prove a finite fermionic Gaussian de Finetti theorem: a k-copy Gaussian state that is n-exchangeable under graded fermionic copy permutations is O(k/n)-close in trace norm to the product of its one-copy marginal. In particular, infinite exchangeability within the Gaussian class implies exact product structure. Third, for spatial fermionic Gaussian states with exponentially clustering mutual information, we prove that the conditional mutual information across a separating buffer decays as O(r−1), and hence, by recoverability, the state admits an O(r−1/2) trace-norm approximation by a recovered state. These results show that product and de Finetti approximations and recoverability estimates for fermionic Gaussian systems can all be derived directly from covariance-level structure.
Understanding the computational complexity of quantum states is a central challenge in quantum many-body physics. In qubit systems, fermionic Gaussian states can be efficiently simulated on classical computers and hence can be employed as a natural baseline for evaluating quantum complexity. In this work, we develop a framework for quantifying fermionic magic resources, also referred to as fermionic non-Gaussianity, which constitutes an essential resource for universal quantum computation. We leverage the algebraic structure of the fermionic commutant to define the fermionic antiflatness (FAF)—an efficiently computable and experimentally accessible measure of non-Gaussianity, with a clear physical interpretation in terms of Majorana fermion correlation functions. Studying systems in equilibrium, we show that FAF detects phase transitions, reveals universal features of critical points, and uncovers special solvable points in many-body systems. Extending the analysis to out-of-equilibrium settings, we demonstrate that fermionic magic resources become more abundant in highly excited eigenstates of many-body systems. We further investigate the growth and saturation of FAF under ergodic many-body dynamics, highlighting the roles of conservation laws and locality in constraining the increase of non-Gaussianity during unitary evolution. This work provides a framework for probing quantum many-body complexity from the perspective of fermionic Gaussian states and opens up directions for investigating fermionic magic resources in many-body systems. Our results establish fermionic non-Gaussianity, alongside entanglement and nonstabilizerness, as a resource relevant not only to foundational studies but also to experimental platforms aiming to achieve quantum advantage.
Fermionic non-Gaussianity, or fermionic magic, is a key resource underlying the computational complexity of fermionic quantum systems, yet tractable and operationally meaningful ways to quantify it remain limited. We address this challenge by developing a convex resource theory of fermionic non-Gaussianity and introducing two families of computable quantities for pure fermionic states, both derived from the Williamson normal form of the covariance matrix. The first family, occupation number entropies, is defined as the Tsallis-α entropy of the occupation numbers. We prove that one member of this family is monotonic under Gaussian protocols, establishing it as a computable convex resource monotone. It consequently lower bounds the number of non-Gaussian gates needed for state preparation. The second family, natural-orbital participation entropies, is given by the Rényi-α entropy of the squared amplitudes of the state in the natural-orbital basis, defined by the eigenvectors of the covariance matrix. They quantify state compressibility in this basis and thus upper bound the classical simulation cost in an orthonormal Gaussian basis. We analyze both families for stabilizer and translation-invariant states, where they simplify and reveal additional structure. We further study representative examples, including random SWAP-doped matchgate circuits and the bond-modulated XXZ model, highlighting the role of non-Gaussianity in many-body phenomena. Our work establishes a resource-theoretic framework for computable fermionic non-Gaussianity that unifies notions arising across quantum information, condensed-matter physics, and quantum chemistry, opening new directions for studying the complexity of quantum many-body systems and providing practical tools to assess the classical simulability of fermionic states relevant for quantum advantage.