  Simulating non-equilibrium phenomena in strongly-interacting quantum
many-body systems, including thermalization, is a promising application of
near-term and future quantum computation. By performing experiments on a
digital quantum computer consisting of fully-connected optically-controlled
trapped ions, we study the role of entanglement in the thermalization dynamics
of a $Z_2$ lattice gauge theory in 2+1 spacetime dimensions. Using
randomized-measurement protocols, we efficiently learn a classical
approximation of non-equilibrium states that yields the gap-ratio distribution
and the spectral form factor of the entanglement Hamiltonian. These observables
exhibit universal early-time signals for quantum chaos, a prerequisite for
thermalization. Our work, therefore, establishes quantum computers as robust
tools for studying universal features of thermalization in complex many-body
systems, including in gauge theories.
