Experimental signatures of Hilbert-space ergodicity: Universal bitstring distributions and applications in noise learning

Systems reaching thermal equilibrium are ubiquitous. For classical systems, this phenomenon is typically understood statistically through ergodicity in phase space, but translating this to quantum systems is a long-standing problem of interest. Recently a strong notion of quantum ergodicity has been proposed, namely that isolated, global quantum states uniformly explore their available state space, dubbed Hilbert-space ergodicity. Here we observe signatures of this process with an experimental Rydberg quantum simulator and various numerical models, before generalizing to the case of a local quantum system interacting with its environment. For a closed system, where the environment is a complementary subsystem, we predict and observe a smooth quantum-to-classical transition in that observables progress from large, quantum fluctuations to small, Gaussian fluctuations as the bath size grows. This transition exhibits universal properties on a quantitative level amongst a wide range of systems, including those at finite temperature, those with itinerant particles, and random circuits. For an open system, where the environment is uncontrolled, we predict the statistics of observables under largely arbitrary noise channels including those with correlated errors, allowing us to discriminate between candidate error models both for continuous Hamiltonian time evolution and for digital random circuits. This allows for computationally efficient experimental noise learning, and more broadly is a new avenue for quantitatively classifying the behavior of noisy quantum systems. Ultimately our results clarify the role of ergodicity in quantum dynamics, with fundamental and practical consequences.