Quantum Ergodicity as the Robustness of Hamiltonian Learning

Quantifying the chaotic nature of quantum systems remains an important open problem in many-body physics. In this work, we introduce a metric for quantum ergodicity based on the robustness of Hamiltonian learning. Our approach offers three key advantages: (i) it not only distinguishes ergodic and non-ergodic systems but also provides a quantitative measure within the ergodic regime; (ii) our approach is robust against imperfections and noise and relies only on local measurements, hence it can be efficiently implemented in real-world experiments; and (iii) it naturally connects to and agrees with established diagnostics of quantum chaos and ergodicity, including eigenstate sensitivity, entanglement properties, and random matrix theory statistics. These features arise from the interplay between the locality of the Hamiltonian and the ergodic nature of its eigenstates. Our findings provide an improved understanding of how existing diagnostics of quantum ergodicity are related and present practical methods for probing them in modern quantum simulators.