Learning a local Hamiltonian from local measurements

TL;DR

This paper demonstrates that local Hamiltonians in quantum systems can be efficiently reconstructed using only local measurements, applicable to eigenstates, Gibbs states, or time-evolved states, with resources scaling linearly with system size.

Contribution

It introduces a method to recover local Hamiltonians from local observables alone, reducing resource requirements and extending applicability to time-dependent Hamiltonians.

Findings

Hamiltonians can be recovered from local measurements in linear resource scaling.

Recovery is possible from eigenstates, Gibbs states, or time-evolved states.

The method's statistical error estimates align with numerical simulations.

Abstract

Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.