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arXiv:2607.02242

Computable measures of fermionic non-Gaussianity from the covariance matrix

Poetri Sonya Tarabunga, Bernhard Jobst, Raúl Morral-Yepes, Marc Langer, Barbara Kraus, Frank Pollmann, Sheng-Hsuan Lin

quant-phcond-mat.str-elSubmitted Jul 2, 2026• Announced Jul 3, 2026• 0 citations
v1 · Jul 2, 2026v2 · Jul 28, 2026
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Abstract

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 measures 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. These measures 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.

Comments
28+28 pages, 7+6 figures

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Fermionic non-Gaussianity, also known as fermionic magic, is a key resource underlying the computational complexity of fermionic quantum systems, yet quantifying it in a computationally tractable and operationally grounded manner remains a central challenge. In this work, we address this gap by establishing a convex resource theory of fermionic non-Gaussianity and introducing two families of computable measures for pure fermionic states, both built from the Williamson normal form of the covariance matrix. The first family, the occupation number entropies, is defined as the Tsallis-α\alphaα entropy of the occupation numbers, obtained as shifted and rescaled Williamson eigenvalues of the covariance matrix. We prove that one member of this family yields a monotone under Gaussian protocols, thereby validating it as a computable convex resource monotone for fermionic non‑Gaussianity. As a consequence, it provides a lower bound on the number of non-Gaussian gates required for state preparation. The second family, the natural-orbital participation entropy, is the R\'enyi-α\alphaα entropy of the squared magnitude of the state's amplitudes in the natural-orbital basis, defined by the eigenvectors of the covariance matrix. Operationally, they quantify the compressibility of the state in the natural-orbital basis, and hence upper bound the classical simulation cost in an orthonormal Gaussian basis. We analyze the two measures in two important settings, for stabilizer and translation-invariant states, where they admit simplified forms and additional structural insights. We investigate both families through several representative examples, including random SWAP-doped matchgate circuits and the bond-modulated XXZ model, showcasing 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.
Fermionic non-Gaussianity is an essential resource for unlocking the full computational power of fermionic quantum platforms. In this work we develop monotones and efficient quantum algorithms for fermionic non-Gaussianity, all built on the eigenvalue structure of the operator Λ=∑j=12nγj⊗γjΛ= \sum_{j=1}^{2n}γ_j\otimesγ_jΛ=∑j=12n​γj​⊗γj​ defined on two copies of an nnn-mode fermionic state, accessible via Bell sampling. In particular, we introduce the \emph{bridge degree} of even pure states, a novel non-Gaussianity monotone defined as the largest eigenvalue of ΛΛΛ whose eigenspace is populated by two copies of the state. Our key technical result is that the bridge degree is non-increasing under post-selected Gaussian protocols, which yields no-go theorems for Gaussian conversion beyond the reach of previously known monotones and shows that the resource theory of fermionic non-Gaussianity is irreversible in the exact-conversion setting. Beyond this, the bridge degree exhibits several further features: it (i) is easy to compute, (ii) is efficiently witnessed through Bell sampling, (iii) lower-bounds the non-Gaussian gate complexity of state preparation, (iv) controls the non-Gaussian gate complexity of producing quantum state designs, and (v) naturally extends to mixed states via the Choi--Jamiołkowski isomorphism. We further develop an approximate variant together with an efficiently measurable lower bound, yielding an experimentally certifiable lower bound on the non-Gaussian cost of approximately preparing any state, based directly on Bell-sampling data. Finally, the same eigenvalue structure underlies two Bell-sampling-based algorithmic primitives, both with polynomial sample complexity: a two-copy Gaussianity test with perfect completeness, optimal among two-copy tests sharing this property, and a test for the state 222-design property of matchgate-invariant ensembles.
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.
Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottleneck by establishing a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, which we prove at Rényi index α=2α=2α=2 for arbitrary subsystem size, and which can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. We apply this framework to characterize fermionic non-local magic across diverse physical regimes: we derive an exact Page-like curve for typical random states, reveal logarithmic scaling at the quantum critical point of the XY model, and establish a quasiparticle picture for magic generation during out-of-equilibrium quantum quenches. Crucially, because our result relies solely on two-point correlation functions, it provides a scalable route for the experimental estimation of fermionic non-local magic in large-scale quantum processors via fermionic shadow tomography.
Fermionic non-Gaussianity is a key resource for universal quantum computation, and its behavior in quantum many-body systems has attracted growing interest. Recently, a convolution-based measure, which we call the magic Rényi entropy (MRE), has been proposed as a resource measure of non-Gaussianity, but its evaluation is limited to small systems due to the computational cost growing exponentially with system size. In this work, we develop a perfect-sampling method to calculate the second-order MRE from matrix product states (MPSs). Our method uses a bounded estimator to control sampling fluctuations and a recursive sweep to draw the samples directly from the input MPS, which allows us to generate each sample with O(nD3)\mathcal O(nD^3)O(nD3) time, where nnn and DDD denote the number of modes and MPS bond dimension, respectively. We benchmark our method on the XXZ chain with up to 128128128 sites, demonstrating that the MRE captures intricate higher-order correlations at large scales, which are inaccessible to covariance-based measures. These results provide a computational tool that quantitatively evaluates many-body fermionic non-Gaussianity by accounting for higher-order correlations.