Optimal testing of fermionic and bosonic Gaussian states
Abstract
We study the problem of property testing quantum states: given a pure state |ψ⟩ that either (A) belongs to some class C of pure states, or (B) is ε-far in trace distance from all states in C, determine which is the case with high probability. It is known that there exists classes C for which the sample complexity of property testing necessarily scales with the size of the system. In this work, we identify classes for which testing only requires a size-independent number of samples. We prove a tight sample complexity of Θ(1/ε2) for the property testing of five important classes of Gaussian quantum states: 1) Fermionic Gaussian states, 2) Slater determinants, 3) bosonic Gaussian states, 4) zero-mean bosonic Gaussian states, and 5) bosonic coherent states. While property testers for some of these classes have been studied previously, those analyses only gave upper bounds scaling polynomially in the number of modes. In each case, our tester is the projection onto the ``top'' irrep of two or three copies. We get to constant sample complexity by lower bounding how quickly the rejection probability increases with distance from the class; the main technical component involves bounding the spectral gap of symmetric extension of the test projector. We then establish optimality by proving matching lower bounds based on binary state discrimination for each class. Our testers and optimality claims also extend to the tolerant setting, wherein ψ may be O(ε)-close to C in case (A), rather than lying exactly inside.
Related
- Testing bosonic and fermionic Gaussian states with a constant number of copies
- Entangled measurements are necessary for optimal tomography of mixed fermionic Gaussian states and of bosonic Gaussian states near the vacuum
- Exponential lower bounds on the fermionic Gaussian rank of magic states and the bosonic coherent state rank of Fock states
- Certifying fermionic Gaussian states (and a little more) with optimal precision dependence
- Robust exponential lower bounds for fermionic and bosonic Gaussian ranks
BibTeX
@misc{iyer2026optimal,
title = {{Optimal testing of fermionic and bosonic Gaussian states}},
author = {Vishnu Iyer and Bryan O'Gorman and Ojas Parekh and Kevin Thompson and Andrew Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39113},
eprint = {2609.39113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39113},
note = {CC0},
}
RIS
TY - UNPB
TI - Optimal testing of fermionic and bosonic Gaussian states
AU - Vishnu Iyer
AU - Bryan O'Gorman
AU - Ojas Parekh
AU - Kevin Thompson
AU - Andrew Zhao
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39113
UR - https://arxiv.org/abs/2609.39113
AB - We study the problem of property testing quantum states: given a pure state |ψ⟩ that either (A) belongs to some class C of pure states, or (B) is ε-far in trace distance from all states in C, determine which is the case with high probability. It is known that there exists classes C for which the sample complexity of property testing necessarily scales with the size of the system. In this work, we identify classes for which testing only requires a size-independent number of samples. We prove a tight sample complexity of Θ(1/ε^2) for the property testing of five important classes of Gaussian quantum states: 1) Fermionic Gaussian states, 2) Slater determinants, 3) bosonic Gaussian states, 4) zero-mean bosonic Gaussian states, and 5) bosonic coherent states. While property testers for some of these classes have been studied previously, those analyses only gave upper bounds scaling polynomially in the number of modes. In each case, our tester is the projection onto the ``top'' irrep of two or three copies. We get to constant sample complexity by lower bounding how quickly the rejection probability increases with distance from the class; the main technical component involves bounding the spectral gap of symmetric extension of the test projector. We then establish optimality by proving matching lower bounds based on binary state discrimination for each class. Our testers and optimality claims also extend to the tolerant setting, wherein ψ may be O(ε)-close to C in case (A), rather than lying exactly inside.
ER -