aiqbench analysis
This research establishes a method to efficiently recover an unknown $n$-qubit Hamiltonian from a single long-time evolution, leveraging classical shadows and random product states. The study proves that for broad families of local Hamiltonians, the target Hamiltonian acts as a unique approximately conserved observable, enabling recovery up to scale. As a corollary, the work provides a weak equilibration statement regarding the decay of infinite-temperature autocorrelations for observables orthogonal to the Hamiltonian.
Learning Hamiltonians at Long Times
arXiv:2606.05690v1 Announce Type: new 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.
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.
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 arXiv-issued DOI via DataCite (pending registration)
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