Near-Optimal Learning of Quantum Many-Body Systems from Gibbs and Thermal Metastable States via the Stationarity Test
Abstract
We study the problem of learning a finite-range quantum Hamiltonian on an n-qubit D-dimensional lattice from copies of its Gibbs states at a known inverse temperature β or its thermal metastable states. Existing efficient algorithms at arbitrary temperature have sample guarantees with an epoly(β) dependence, leaving open whether the exponent can be made linear in β. We achieve this improvement: for β≥1, our algorithm learns all Hamiltonian coefficients to ℓ∞-error ε with probability at least 1−δ using tO(eO(β)ε−2log(n/δ)) copies and total time tO(n eO(β)ε−2log(n/δ)). The measurements act on individual copies and are supported on balls of radius O(β2+βlog(1/ε)). Our sample complexity achieves the optimal exponential order in β, with optimal ε−2 dependence up to logarithmic factors. Building on the stationarity-test framework \citep{Bergamaschi2026Stationarity}, we prove a unified Fisher-information coercivity bound showing that errors in the candidate Hamiltonian produce a detectable signal with only an e−O(β) loss in sensitivity. The key is a smooth frequency filter whose bandwidth remains independent of temperature, together with sharper estimates relating this signal to the Hamiltonian coefficients. For thermal metastable states, the guarantee holds under local drift ζ≤ e−Cβ2D+1, for a geometry-dependent constant C>0, and above an accuracy floor tO(eO(β)ζ). The argument applies to both Gibbs and thermal metastable states and, combined with local measurements and projected stochastic gradient descent (SGD), yields both learning guarantees.
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BibTeX
@misc{yang2026nearoptimal,
title = {{Near-Optimal Learning of Quantum Many-Body Systems from Gibbs and Thermal Metastable States via the Stationarity Test}},
author = {Tai Hsuan Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01520},
eprint = {2611.01520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01520},
}
RIS
TY - UNPB
TI - Near-Optimal Learning of Quantum Many-Body Systems from Gibbs and Thermal Metastable States via the Stationarity Test
AU - Tai Hsuan Yang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01520
UR - https://arxiv.org/abs/2611.01520
AB - We study the problem of learning a finite-range quantum Hamiltonian on an n-qubit D-dimensional lattice from copies of its Gibbs states at a known inverse temperature β or its thermal metastable states. Existing efficient algorithms at arbitrary temperature have sample guarantees with an e^poly(β) dependence, leaving open whether the exponent can be made linear in β. We achieve this improvement: for β≥1, our algorithm learns all Hamiltonian coefficients to ℓ_∞-error ε with probability at least 1−δ using tO(e^O(β)ε^−2log(n/δ)) copies and total time tO(n e^O(β)ε^−2log(n/δ)). The measurements act on individual copies and are supported on balls of radius O(β^2+βlog(1/ε)). Our sample complexity achieves the optimal exponential order in β, with optimal ε^−2 dependence up to logarithmic factors. Building on the stationarity-test framework \citep{Bergamaschi2026Stationarity}, we prove a unified Fisher-information coercivity bound showing that errors in the candidate Hamiltonian produce a detectable signal with only an e^−O(β) loss in sensitivity. The key is a smooth frequency filter whose bandwidth remains independent of temperature, together with sharper estimates relating this signal to the Hamiltonian coefficients. For thermal metastable states, the guarantee holds under local drift ζ≤ e^−Cβ^2D+1, for a geometry-dependent constant C>0, and above an accuracy floor tO(e^O(β)ζ). The argument applies to both Gibbs and thermal metastable states and, combined with local measurements and projected stochastic gradient descent (SGD), yields both learning guarantees.
ER -