Fermionic non-Gaussianity via Bell sampling: monotones and efficient quantum algorithms

@misc{tarabunga2026fermionic,
  title = {{Fermionic non-Gaussianity via Bell sampling: monotones and efficient quantum algorithms}},
  author = {Poetri Sonya Tarabunga},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2606.05066},
  eprint = {2606.05066},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.05066},
}

TY  - UNPB
TI  - Fermionic non-Gaussianity via Bell sampling: monotones and efficient quantum algorithms
AU  - Poetri Sonya Tarabunga
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2606.05066
UR  - https://arxiv.org/abs/2606.05066
AB  - 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=1^2nγ_j⊗γ_j defined on two copies of an n-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 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.
ER  -