Physics-inspired Potts machines of coupled oscillators for low-energy sampling and combinatorial optimization

Sampling low-energy configurations of the standard q-state Potts model is essential in the study of phase transitions, critical phenomena, complex systems, and combinatorial optimization, but remains challenging. While physics-inspired dynamical sampling has been extensively explored for the Ising case (q = 2) in the form of Ising machines, direct dynamical system frameworks for the standard q-state Potts model have received less attention. We propose a class of physics-inspired dynamical samplers, referred to as oscillator Potts machines (OPMs), that directly target the standard q-state Potts model. We show that the OPM exhibits a systematic low-energy bias with respect to the underlying Potts energy landscape. We demonstrate that the OPM can be realized in a network of self-sustaining oscillators, indicating that it is compatible with standard hardware technologies. We apply OPMs to max-k-cut problems by mapping these instances to the Potts Hamiltonians. These results position the OPM as a physically grounded dynamical system framework for multistate sampling and combinatorial optimization. Many optimization and sampling problems involve variables with more than two states, beyond the binary Ising model. For the standard q-state Potts model, we introduce an oscillator-based dynamical system that favors low-energy states and can be implemented with coupled oscillators.