aiqbench analysis
The paper gives a theoretical improvement to the sample complexity of learning local Hamiltonians from Gibbs and thermal metastable states, reducing the dependence on inverse temperature from $e^{\mathrm{poly}(\beta)}$ to $e^{O(\beta)}$. The result is a proof-level advance in quantum learning theory, and its practical relevance depends on whether the stated constants and the $\beta \ge 1$ regime hold up in experimental settings. Readers following quantum characterization and learning should watch for follow-up empirical work and whether the stationarity-test framework extends to other state families.
Near-Optimal Learning of Quantum Many-Body Systems from Gibbs and Thermal Metastable States via the Stationarity Test
arXiv:2611.01520v1 Announce Type: new 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 $\beta$ or its thermal metastable states. Existing efficient algorithms at arbitrary temperature have sample guarantees with an $e^{\poly(\beta)}$ dependence, leaving open whether the exponent can be made linear in $\beta$. We achieve this improvement: for $\beta\ge1$, our algorithm learns all Hamiltonian coefficients to $\ell_\infty$-error $\epsilon$ with probability at least $1-\delta$ using $\tO\bigl(e^{O(\beta)}\epsilon^{-2}\log(n/\delta)\bigr)$ copies and total time $\tO\bigl(n e^{O(\beta)}\epsilon^{-2}\log(n/\delta)\bigr)$. The measurements act on individual copies and are supported on balls of radius $O\bigl(\beta^2+\beta\log(1/\epsilon)\bigr)$. Our sample complexity achieves the optimal exponential order in $\beta$, with optimal $\epsilon^{-2}$ 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 $e^{-O(\beta)}$ 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 $\zeta\le e^{-C\beta^{2D+1}}$, for a geometry-dependent constant $C>0$, and above an accuracy floor $\tO(e^{O(\beta)}\zeta)$. 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.
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 $\beta$ or its thermal metastable states. Existing efficient algorithms at arbitrary temperature have sample guarantees with an $e^{\poly(\beta)}$ dependence, leaving open whether the exponent can be made linear in $\beta$. We achieve this improvement: for $\beta\ge1$, our algorithm learns all Hamiltonian coefficients to $\ell_\infty$-error $\epsilon$ with probability at least $1-\delta$ using $\tO\bigl(e^{O(\beta)}\epsilon^{-2}\log(n/\delta)\bigr)$ copies and total time $\tO\bigl(n e^{O(\beta)}\epsilon^{-2}\log(n/\delta)\bigr)$. The measurements act on individual copies and are supported on balls of radius $O\bigl(\beta^2+\beta\log(1/\epsilon)\bigr)$. Our sample complexity achieves the optimal exponential order in $\beta$, with optimal $\epsilon^{-2}$ 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 $e^{-O(\beta)}$ 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 $\zeta\le e^{-C\beta^{2D+1}}$, for a geometry-dependent constant $C>0$, and above an accuracy floor $\tO(e^{O(\beta)}\zeta)$. 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.
arXiv:2611.01520 [quant-ph]
(or arXiv:2611.01520v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2611.01520 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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