  An optical-lattice quantum simulator is an ideal experimental platform to
investigate non-equilibrium dynamics of a quantum many-body system, which is in
general hard to simulate with classical computers. Here, we use our quantum
simulator of the Bose-Hubbard model to study dynamics far from equilibrium
after a quantum quench. We successfully confirm the energy conservation law in
the one- and three-dimensional systems and extract the propagation velocity of
the single-particle correlation in the one- and two-dimensional systems. We
corroborate the validity of our quantum simulator through quantitative
comparisons between the experiments and the exact numerical calculations in one
dimension. In the computationally hard cases of two or three dimensions, by
using the quantum-simulation results as references, we examine the performance
of a numerical method, namely the truncated Wigner approximation, revealing its
usefulness and limitation. This work constitutes an exemplary case for the
usage of analog quantum simulators.
