  Ergodicity and chaos play an integral role in the dynamical behavior of
many-particle systems and are crucial to the formulation of statistical
mechanics. Still, a general understanding of how randomness and chaos emerge in
the dynamical evolution of closed quantum systems remains elusive. Here, we
develop an experimental platform for the realization of canonical quantum
chaotic Hamiltonians based on quantum simulation with synthetic lattices. We
map the angular momentum projection states of an effective quantum spin onto
the linear momentum states of a $^{87}$Rb Bose-Einstein condensate, which can
alternatively be viewed as lattice sites in a synthetic dimension. This
synthetic lattice, with local and dynamical control of tight-binding lattice
parameters, enables new capabilities related to the experimental study of
quantum chaos. In particular, the capabilities of our system let us tune the
effective size of our spin, allowing us to illustrate how classical chaos can
emerge from a discrete quantum system. Moreover, spectroscopic control over our
synthetic lattice allows us to explore unique aspects of our spin's dynamics by
measuring the out-of-time-ordered correlation function, and enables future
investigations into entirely new classes of chaotic systems.
