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

TL;DR

This paper demonstrates a scalable, reconfigurable photonic platform capable of implementing quantum walks on lattices with various topologies, including cyclic, cylindrical, and toroidal, enabling the study of topological effects and dynamics.

Contribution

The authors introduce a novel photonic platform that allows programmable quantum walks on lattices with periodic boundary conditions and topologies, including a method for controlling topology via reciprocal space sampling.

Findings

Realization of quantum walks on 1D cyclic and 2D cylindrical/toroidal lattices.

Observation of wavepacket refocusing, breathing modes, and topology-dependent trajectories.

Mapping 2D walks on a cylinder to 1D walks with high-dimensional coins.

Abstract

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, such as cyclic or toroidal geometries. Here, we employ a recently developed platform that enables the implementation of arbitrary translationally invariant unitary operations on one- and two-dimensional lattices, and demonstrate a natural mechanism for introducing periodic boundary conditions. Our approach leverages direct access to the reciprocal lattice, where discrete sampling of the unitary evolution effectively enforces the desired topology. We program our platform to realize quantum walks on 1D cyclic lattices and 2D lattices with cylindrical or toroidal topologies. The lattice size can be readily tuned by adjusting the sampling density in reciprocal space. By controlling reciprocal-space occupancy, 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. We further demonstrate a form of dimensional reduction by mapping a 2D quantum walk on a cylinder to a 1D walk with a high-dimensional coin. These results establish a versatile platform for realizing a broad class of optical mode transformations within bounded Hilbert spaces.