  Statistical mechanics is founded on the assumption that all accessible
configurations of a system are equally likely. This requires dynamics that
explore all states over time, known as ergodic dynamics. In isolated quantum
systems, however, the occurrence of ergodic behavior has remained an
outstanding question. Here, we demonstrate ergodic dynamics in a small quantum
system consisting of only three superconducting qubits. The qubits undergo a
sequence of rotations and interactions and we measure the evolution of the
density matrix. Maps of the entanglement entropy show that the full system can
act like a reservoir for individual qubits, increasing their entropy through
entanglement. Surprisingly, these maps bear a strong resemblance to the phase
space dynamics in the classical limit; classically chaotic motion coincides
with higher entanglement entropy. We further show that in regions of high
entropy the full multi-qubit system undergoes ergodic dynamics. Our work
illustrates how controllable quantum systems can investigate fundamental
questions in non-equilibrium thermodynamics.
