Learning Hamiltonians at Long Times

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 $\frac{1}{2^n}\Vert [U(t),A]{\Vert }_F^2\ge 1/\text{poly}(n)$. 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.

11+54 pages, 5 figures