Practical Tests and Witnesses of Fermionic non-Gaussianity

Detecting when a quantum state leaves the efficiently simulable fermionic Gaussian regime is a central task for benchmarking quantum devices and certifying fermionic magic resources. We develop practical tests and witnesses based on fermionic antiflatness (FAF), a covariance-matrix-based measure of non-Gaussianity. For $n$-qubit states, we estimate FAF using two complementary protocols: two-copy Bell measurements and a single-copy scheme based on commuting matchings of Majorana bilinears. These yield testers that distinguish pure Gaussian states from states $\epsilon$-far from the Gaussian set, using $O(n^2/\epsilon^2)$ two-copy Bell measurements or $O(n^3/\epsilon^4)$ single-copy measurements, improving the state of the art in the dependence on both $n$ and $\epsilon$. For mixed states, we introduce a purity-corrected FAF witness that certifies non-Gaussianity and is highly robust to noise. With our witness, we demonstrate on the IQM quantum computer that noise can both reduce and enhance non-Gaussianity. Finally, by examining pseudo non-Gaussianity, we show that the cryptographic task of pseudorandom-state generation requires extensive fermionic non-Gaussianity. Together, these results provide experimentally accessible tools for detecting, witnessing, and quantifying non-Gaussian fermionic resources.