[2607.02242] Computable fermionic non-Gaussianity from the covariance matrix

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Summary:Fermionic non-Gaussianity, or fermionic magic, is a key useful resource underlying the computational complexity of fermionic quantum programs, but tractable and operationally significant methods to quantify it stay restricted. We deal with this problem by creating a convex useful resource idea of fermionic non-Gaussianity and introducing two households of computable portions for pure fermionic states, each derived from the Williamson regular type of the covariance matrix. The primary household, occupation quantity entropies, is outlined because the Tsallis-$alpha$ entropy of the occupation numbers. We show that one member of this household is monotonic below Gaussian protocols, establishing it as a computable convex useful resource monotone. It consequently decrease bounds the variety of non-Gaussian gates wanted for state preparation. The second household, natural-orbital participation entropies, is given by the Rényi-$alpha$ entropy of the squared amplitudes of the state within the natural-orbital foundation, outlined by the eigenvectors of the covariance matrix. They quantify state compressibility on this foundation and thus higher sure the classical simulation price in an orthonormal Gaussian foundation. We analyze each households for stabilizer and translation-invariant states, the place they simplify and reveal further construction. We additional examine consultant examples, together with random SWAP-doped matchgate circuits and the bond-modulated XXZ mannequin, highlighting the position of non-Gaussianity in many-body phenomena. Our work establishes a resource-theoretic framework for computable fermionic non-Gaussianity that unifies notions arising throughout quantum data, condensed-matter physics, and quantum chemistry, opening new instructions for learning the complexity of quantum many-body programs and offering sensible instruments to evaluate the classical simulability of fermionic states related for quantum benefit.