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Fermionic Non-Gaussianity via Bell Sampling: Monotones and Efficient Quantum Algorithms

Poetri Sonya Tarabunga*

  • *Contact author: poetri.tarabunga@tum.de
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PRX Quantum 7, 033065 – Published 23 September, 2026

DOI: https://doi.org/10.1103/1bkz-7wf2

Abstract

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⊗γj defined on two copies of an n-mode fermionic state, accessible via Bell sampling. In particular, we introduce the bridge degree of even pure states, a novel non-Gaussianity monotone defined as the largest eigenvalue of Λ 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 2-design property of matchgate-invariant ensembles.

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Popular Summary

Quantum computers built from fermions—particles like electrons—are not always more powerful than classical ones, since many of their natural operations can be simulated classically. The ingredient for classical hardness is a resource called non-Gaussianity, which has been difficult to measure. We introduce a simple way to quantify it, called the bridge degree. It can be easily calculated on quantum states, and also probed in the laboratory by measuring two copies of a state at once. This allows us to prove that certain conversions between quantum states are impossible, and that the resource is lost for good once spent: you can never fully recover it by converting back. It also gives experimenters a practical way to check how much of the resource their hardware actually produces. Investigating this resource is highly useful to map out what fermionic quantum machines will need in order to beat classical computers.

Article Text

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PRX Quantum 7, 033065– Published 23 September, 2026
  • Received 10 June 2026
  • Revised 7 August 2026
  • Accepted 19 August 2026
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