  Synthetic dimensions have generated great interest for studying many types of
topological, quantum, and many-body physics, and they offer a flexible platform
for simulation of interesting physical systems, especially in high dimensions.
In this Letter, we describe a programmable photonic device capable of emulating
the dynamics of a broad class of Hamiltonians in lattices with arbitrary
topologies and dimensions. We derive a correspondence between the physics of
the device and the Hamiltonians of interest, and we simulate the physics of the
device to observe a wide variety of physical phenomena, including chiral states
in a Hall ladder, effective gauge potentials, and oscillations in
high-dimensional lattices. Our proposed device opens new possibilities for
studying topological and many-body physics in near-term experimental platforms.
