Abstract
Certain ensembles of random unitaries or states – those forming approximate designs – provide useful tools in quantum information theory, enabling a wide range of applications from the analysis of the circuit complexity to the development of benchmarking protocols. In this talk, we report an unexpected discovery inspired by modern quantum simulation experiments: natural quantum many-body dynamics give rise to pure state ensembles with universal statistical properties. Specifically, we consider two types of ensembles: (i) the projected ensemble is formed by random pure states for a subsystem obtained by performing projective measurement on its complement, and (ii) the temporal ensemble is obtained by global quantum states evolved under ergodic Hamiltonian for different time durations. In both cases, the ensembles display universal statistical properties, a phenomenon we dub deep thermalization and Hilbert-space ergodicity, respectively. The special case of infinite effective temperature corresponds to the emergence of approximate state designs. We discuss a number of analytic and numerical results supporting our claim as well as how to observe and utilize the universal properties in modern experiments. These findings allow us to generalize the linear-cross entropy benchmark (XEB) to analog quantum simulators, to develop further applications such as parameter estimation and noise characterization and could establish quantifiable connections between random matrix theory and realistic ergodic quantum many-body dynamics.