Learning a Local Hamiltonian from Local Measurements

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.

Learning the dynamics of open quantum systems from their steady states

“…The recovery error Δ we find in figure 1(a) is slightly higher (by a factor of »1.25) than the estimate of equation (8), derived in [40]. In contrast to our results in this work, the recovery error obtained in [40] was lower than the prediction of the same estimate, which is indeed expected to be pessimistic due to the use of Jensen's inequality.…”

“…We believe the difference is due to the different noise model used in both papers: here we add noise to each measured observable, while in [40] we added independent noise to each of the entries of K (even when they contain the same observable). This is because in [40], we wished to test the theoretical validity of the error estimate. The estimate assumes that the noise in each entry of the constraint matrix K is independent, and we thus added an independent random noise to each of its entries.…”

“…Using perturbation theory, we estimated in [40] the reconstruction error due to independent random noise with standard deviation ò added to each element of K where l m are the eigenvalues of K K T 5 (i.e. the squared singular values of K ).…”

“…An isolated quantum system can be characterized by learning its underlying Hamiltonian. This can be achieved by monitoring the dynamics that the Hamiltonian generates [18][19][20][21][22][23][24][25][26][27][28][29][30][31], or by measuring local observables in one of its eigenstates or thermal states [32][33][34][35][36][37][38][39][40]. However, realistic quantum systems are never fully isolated.…”

“…In this work, we study this question by providing an efficient algorithm for learning the dynamics of local Lindbladians from their steady states. Extending the methods of [40], our algorithm exploits strong constraints that locality imprints on the steady states of generic local Lindbladians. Using this algorithm, we (i) explore which types of Lindbladians can be accurately reconstructed from their steady states, (ii) study numerically and analytically the system-size scaling of the reconstruction accuracy, and (iii) show that coupling to a bath can in fact facilitate the reconstruction of certain classes of Hamiltonians, which pose a challenge for methods based on their eigenstates or Gibbs states.…”