Hamiltonian and Liouvillian learning in weakly-dissipative quantum many-body systems

TL;DR

This paper explores methods for learning Hamiltonian and Liouvillian operators in weakly-dissipative quantum many-body systems using non-equilibrium dynamics, emphasizing ansatz parametrization and classical post-processing to improve efficiency and identify system parameters.

Contribution

It introduces and compares new strategies for Hamiltonian and Liouvillian learning that utilize parametrization and classical post-processing, enhancing the identification of relevant system parameters.

Findings

Learning error decreases with the inverse square root of the number of experimental runs.

Learning error plateaus, indicating the presence of missing ansatz terms.

Parametrization reduces complexity and helps identify relevant parameters.

Abstract

We discuss Hamiltonian and Liouvillian learning for analog quantum simulation from non-equilibrium quench dynamics in the limit of weakly dissipative many-body systems. We present and compare various methods and strategies to learn the operator content of the Hamiltonian and the Lindblad operators of the Liouvillian. We compare different ans\"atze based on an experimentally accessible "learning error" which we consider as a function of the number of runs of the experiment. Initially, the learning error decreases with the inverse square root of the number of runs, as the error in the reconstructed parameters is dominated by shot noise. Eventually the learning error remains constant, allowing us to recognize missing ansatz terms. A central aspect of our approaches is to (re-)parametrize ans\"atze by introducing and varying the dependencies between parameters. This allows us to identify the relevant parameters of the system, thereby reducing the complexity of the learning task. Importantly, this (re-)parametrization relies solely on classical post-processing, which is compelling given the finite amount of data available from experiments. We illustrate and compare our methods with two experimentally relevant spin models.