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Fermionic Magic Resources of Quantum Many-Body Systems

Piotr Sierant1,*, Paolo Stornati1,†, and Xhek Turkeshi2,‡

  • *Contact author: piotr.sierant@bsc.es
  • †Contact author: paolo.stornati@bsc.es
  • ‡Contact author: turkeshi@thp.uni-koeln.de
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PRX Quantum 7, 010302 – Published 5 January, 2026

DOI: https://doi.org/10.1103/3yx4-1j27

Abstract

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.

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Popular Summary

Quantum computers promise to harness the laws of quantum mechanics to solve problems beyond the reach of classical computers, accelerating progress across science and technology. However, this computational advantage requires quantum states to effectively explore the vast many-body Hilbert space in ways that classical machines cannot efficiently simulate. How can we determine whether a given quantum state admits an efficient classical description or genuinely provides quantum speedup?

In this work, we address this fundamental challenge by focusing on fermionic Gaussian states, which underpin numerous physical phenomena—from the periodic table and free-electron band theory to degenerate Fermi gases in neutron stars—while remaining efficiently simulable on classical computers. By treating the “distance” from the manifold of Gaussian states as a quantifiable resource-fermionic non-Gaussianity, we establish a unified resource-theoretic framework to diagnose when, and to what extent, a quantum state evades classical simulability.

We develop a family of fermionic non-Gaussianity measures grounded in the algebraic structure of Gaussian states, demonstrating that these measures can be computed efficiently in numerical simulations and directly determined in experiments. We validate our approach using analytically solvable examples and illustrate its utility in both equilibrium and nonequilibrium many-body scenarios. This framework not only uncovers hidden structures in quantum states that obstruct quantum advantage but also provides insights into the intrinsic quantumness of many-body states.

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PRX Quantum 7, 010302– Published 5 January, 2026
  • Received 12 June 2025
  • Revised 6 November 2025
  • Accepted 17 November 2025
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