  Quantum interference can terminate energy growth in a continually kicked
system, via a single-particle ergodicity-breaking mechanism known as dynamical
localization. The effect of many-body interactions on dynamically localized
states, while important to a fundamental understanding of quantum decoherence,
has remained unexplored despite a quarter-century of experimental studies. We
report the experimental realization of a tunably-interacting kicked quantum
rotor ensemble using a Bose-Einstein condensate in a pulsed optical lattice. We
observe signatures of a prethermal localized plateau, followed for interacting
samples by interaction-induced anomalous diffusion with an exponent near one
half. Echo-type time reversal experiments establish the role of interactions in
destroying reversibility. These results quantitatively elucidate the dynamical
transition to many-body quantum chaos, advance our understanding of quantum
anomalous diffusion, and delimit some possibilities for protecting quantum
information in interacting driven systems.

Quantum Gases;Statistical Mechanics;Quantum Physics