Learning Hamiltonians at Long Times

• URL: http://arxiv.org/abs/2606.05690v1
• Date: Thu, 04 Jun 2026 04:16:12 GMT
• Title: Learning Hamiltonians at Long Times
• Abstract summary: We study the problem of learning an unknown $n$-qubit Hamiltonian $H$ from $U = e-iHt$, where $t$ may be arbitrarily large.<n>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 normalized to $H$ satisfies $tfrac12n|[U(t),A]|_F2 geq 1/textpoly(n)$
• Score: 0.45943788817562164
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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