Certifying fermionic Gaussian states (and a little more) with optimal precision dependence
Abstract
Fermionic Gaussian states are the workhorse reference states for qubit-based quantum simulation of fermionic systems, yet existing certification protocols either proceed via fidelity estimation, leading to suboptimal sample complexity in the target precision ε, only apply to Haar-typical states, or require adaptive measurements on a large fraction of qubits. We give an adaptive protocol that certifies any d-mode pure fermionic Gaussian state using O(d2ε−1) copies, single-qubit measurements with only one qubit measured adaptively per copy, and O(d3) classical processing time per copy. The ε−1 dependence is optimal for the certification problem, even among strategies using entangled measurements. The sample complexity is controlled by the spectral gap of a d↔ d−2 down-up walk---a Markov chain studied in the theory of high-dimensional expanders, equivalent to a two-site Glauber dynamics in statistical physics---relating certification efficiency to relaxation time to equilibrium. The worst-case bound is tight for this protocol and is attained by physically relevant states, including ground states of the fully dimerized Su-Schrieffer-Heeger (SSH) chain and BCS pair states. In contrast, numerics up to d=14 suggest that O(d ε−1) copies suffice to certify Haar-typical Gaussian states, a factor of d improvement over the worst-case bound. Finally, because the bound depends only on the target's computational-basis distribution, the protocol extends to efficiently phase-dressed states---states obtained by injecting an efficiently computable diagonal phase to Gaussian states---with the same sample complexity. The class includes a continuous family of four-mode non-Gaussian magic states for matchgate computation.
Related
- Entangled measurements are necessary for optimal tomography of mixed fermionic Gaussian states and of bosonic Gaussian states near the vacuum
- Optimal testing of fermionic and bosonic Gaussian states
- Improved classical optimization of fermionic Hamiltonians beyond Gaussian states
- Weakly interacting fermionic Gibbs states are Gaussian mixtures and classically simulable
- Operator-Space Entanglement Entropy in Random Fermionic Gaussian Dynamics
BibTeX
@misc{dangniam2026certifying,
title = {{Certifying fermionic Gaussian states (and a little more) with optimal precision dependence}},
author = {Ninnat Dangniam and Laphas Premcharoen and Metrasit Sripech and Thiparat Chotibut},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28233},
eprint = {2609.28233},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28233},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Certifying fermionic Gaussian states (and a little more) with optimal precision dependence
AU - Ninnat Dangniam
AU - Laphas Premcharoen
AU - Metrasit Sripech
AU - Thiparat Chotibut
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28233
UR - https://arxiv.org/abs/2609.28233
AB - Fermionic Gaussian states are the workhorse reference states for qubit-based quantum simulation of fermionic systems, yet existing certification protocols either proceed via fidelity estimation, leading to suboptimal sample complexity in the target precision ε, only apply to Haar-typical states, or require adaptive measurements on a large fraction of qubits. We give an adaptive protocol that certifies any d-mode pure fermionic Gaussian state using O(d^2ε^−1) copies, single-qubit measurements with only one qubit measured adaptively per copy, and O(d^3) classical processing time per copy. The ε^−1 dependence is optimal for the certification problem, even among strategies using entangled measurements. The sample complexity is controlled by the spectral gap of a d↔ d−2 down-up walk---a Markov chain studied in the theory of high-dimensional expanders, equivalent to a two-site Glauber dynamics in statistical physics---relating certification efficiency to relaxation time to equilibrium. The worst-case bound is tight for this protocol and is attained by physically relevant states, including ground states of the fully dimerized Su-Schrieffer-Heeger (SSH) chain and BCS pair states. In contrast, numerics up to d=14 suggest that O(d ε^−1) copies suffice to certify Haar-typical Gaussian states, a factor of d improvement over the worst-case bound. Finally, because the bound depends only on the target's computational-basis distribution, the protocol extends to efficiently phase-dressed states---states obtained by injecting an efficiently computable diagonal phase to Gaussian states---with the same sample complexity. The class includes a continuous family of four-mode non-Gaussian magic states for matchgate computation.
ER -