Learning Hamiltonians at Long Times

Quantum Physics [Submitted on 4 Jun 2026] Learning Hamiltonians at Long Times Constantin Cedillo Vayson de Pradenne, Jordan Cotler, Hsin-Yuan Huang We study the problem of learning an unknown n-qubit Hamiltonian H from U = e^{-iHt} for a single time t, where t may be arbitrarily large. For broad families of local Hamiltonians, we prove that, with high probability over H and t, any sum of local observables A that is normalized and orthogonal to H satisfies \tfrac{1}{2^n}\|[U(t),A]\|_F^2 \geq 1/\text{poly}(n). The Hamiltonian is therefore the unique approximately conserved local observable, and we can efficiently recover H, up to scale, as the approximate null vector of a data matrix built from random product-state inputs and classical shadows. As a corollary, we obtain a weak equilibration statement: the infinite-temperature autocorrelation of every sum of local observables orthogonal to H decays by at least an inverse-polynomial amount. Comments: 11+54 pages, 5 figures Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2606.05690 [quant-ph]   (or arXiv:2606.05690v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2606.05690 Focus to learn more Submission history From: Constantin Cedillo Vayson De Pradenne [view email] [v1] Thu, 4 Jun 2026 04:16:12 UTC (1,557 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-06 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)