Programmable photonic quantum walks on lattices with cyclic, toroidal, and cylindrical topology

Photonic implementations of unitary processes on lattice structures, such as quantum walks, have been demonstrated across various architectures. However, few platforms offer the combined advantages of scalability, reconfigurability, and the ability to simulate dynamics on lattices with periodic boundary conditions. Here, we employ a recently developed platform that enables the implementation of arbitrary translation-invariant unitary operations on one- and two-dimensional lattices, and demonstrate a mechanism for introducing periodic boundary conditions. Our approach leverages the possibility of shaping the evolution operator directly in the reciprocal lattice, where discrete sampling of the unitary effectively enforces the desired topology. We program our platform to realize quantum walks on one-dimensional cyclic lattices and two-dimensional lattices with cylindrical or toroidal topologies. The lattice size is controlled by adjusting the sampling density in reciprocal space. We investigate the dynamics of localized states and wavepackets, observing refocusing behavior, breathing modes modulated by reciprocal-space discretizations, and wavepacket trajectories that reflect the underlying topology. Finally, we demonstrate a form of dimensional reduction by mapping a two-dimensional quantum walk on a cylinder to a one-dimensional walk with a high-dimensional coin. These results establish a novel paradigm for realizing a broad class of optical mode transformations within bounded Hilbert spaces.