Universal fluctuations and noise learning from Hilbert-space ergodicity

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 quantum notion of 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 is universal on a quantitative level amongst a wide range of systems, including those at finite temperature, those with itinerant particles, and random circuits. Then, we consider the case of an open system interacting noisily with an external environment. We predict the statistics of observables under largely arbitrary noise channels including those with correlated errors, allowing us to discriminate candidate error models both for continuous Hamiltonian time evolution and for digital random circuits. Ultimately our results clarify the role of ergodicity in quantum dynamics, with fundamental and practical consequences.