Testing quantum Gaussianity with constant sample complexity
Abstract
Efficiently testing whether a quantum state possesses a given structure is both a fundamental and a practical task in quantum information. Among the most important structured families are Gaussian states, which underpin quantum optics and many-body physics while defining paradigmatic regimes of efficient classical simulation. Despite the central role of Gaussian states, optimal tests of Gaussianity have remained unknown. Here we give optimal and robust algorithms for testing fermionic and bosonic Gaussian states, with sample complexity independent of system size, particle number, and energy. Remarkably, the protocols implement experimentally feasible measurements associated with known exact symmetry characterizations of Gaussianity, using Bell sampling on two copies for fermions and passive interferometry with photon counting on three copies for bosons. We further obtain tolerant testers requiring joint measurements on a larger, but still constant, number of copies. At the heart of our analysis is a unified approach---which we call the gradient-flow method---that applies across all structured families considered here and underlies our optimal testers: representation theory turns approximate satisfaction of the defining symmetry into quantitative control of the gradient of the rejection probability, allowing a continuous descent from the input state to the target family whose length bounds their distance.
Related
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- An Exponential Sample-Complexity Advantage for Coherent Quantum Inference
BibTeX
@misc{rad2026testing,
title = {{Testing quantum Gaussianity with constant sample complexity}},
author = {Mahtab Yaghubi Rad and Ricard Puig and Saksham Hassanandani and Luke Coffman and Jens Eisert and Carlos Bravo-Prieto and Antonio Anna Mele},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40270},
eprint = {2609.40270},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40270},
}
RIS
TY - UNPB
TI - Testing quantum Gaussianity with constant sample complexity
AU - Mahtab Yaghubi Rad
AU - Ricard Puig
AU - Saksham Hassanandani
AU - Luke Coffman
AU - Jens Eisert
AU - Carlos Bravo-Prieto
AU - Antonio Anna Mele
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40270
UR - https://arxiv.org/abs/2609.40270
AB - Efficiently testing whether a quantum state possesses a given structure is both a fundamental and a practical task in quantum information. Among the most important structured families are Gaussian states, which underpin quantum optics and many-body physics while defining paradigmatic regimes of efficient classical simulation. Despite the central role of Gaussian states, optimal tests of Gaussianity have remained unknown. Here we give optimal and robust algorithms for testing fermionic and bosonic Gaussian states, with sample complexity independent of system size, particle number, and energy. Remarkably, the protocols implement experimentally feasible measurements associated with known exact symmetry characterizations of Gaussianity, using Bell sampling on two copies for fermions and passive interferometry with photon counting on three copies for bosons. We further obtain tolerant testers requiring joint measurements on a larger, but still constant, number of copies. At the heart of our analysis is a unified approach---which we call the gradient-flow method---that applies across all structured families considered here and underlies our optimal testers: representation theory turns approximate satisfaction of the defining symmetry into quantitative control of the gradient of the rejection probability, allowing a continuous descent from the input state to the target family whose length bounds their distance.
ER -