  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.
