Solving the Planar Potts Problem with Photonics-Inspired Clock Models

Combinatorial optimization problems involve optimizing a cost function defined over all the combinations of a discrete set of objects. Typically, the size of the configuration space of such problems grows faster than polynomial scaling with the number of variables involved, which makes an exhaustive search for the optimal combination impractical. Thus, there is a great interest in developing methods for finding an approximation of the global optimal solution with polynomial scaling. Here, inspired by a recently proposed photonic Potts machine, we introduce a simplified nonlinear dynamical system that can faithfully model the planar Potts Hamiltonian. By employing two annealing mechanisms based on adiabatic parametric deformations of the governing Lyapunov function and harnessing chaotic dynamics, we can dynamically search for the ground state of the Potts Hamiltonian and find high-quality approximate solutions to the underlying NP-hard combinatorial problem. The simplicity and the inherently parallel nature of this system of coupled equations allow for its fast and efficient numerical integration on high-performance digital processors.