From Ising to Potts: Physics-inspired Potts machines of coupled oscillators for low-energy sampling and combinatorial optimization

TL;DR

The paper addresses the challenge of sampling low-energy configurations in the $q$-state Potts model by introducing the oscillator Potts machine (OPM), a physically inspired sampler that directly targets multi-state Potts energies via an equilibrium-preserving relaxation and overdamped Langevin dynamics. It provides a theoretical link between continuous-phase dynamics and discrete Potts configurations, proves the existence of structurally stable sampling points, and demonstrates hardware realizability through a CMOS ring-oscillator circuit, including a 3-state proof-of-concept. Empirical results show a quantifiable low-energy bias after quantization and competitive performance on both small Potts instances and large-scale max-$K$-cut benchmarks, often surpassing other physics-inspired or heuristic approaches. Together, these results position the OPM as a scalable, hardware-friendly framework for multi-state sampling and combinatorial optimization with potential applications in associative memory and beyond.

Abstract

The $q$-state Potts model is a fundamental model in statistical physics that generalizes the Ising model and plays a key role in the study of phase transitions, critical phenomena, complex systems, and combinatorial optimization. Sampling low-energy configurations of the $q$-state Potts model is essential to these studies, but it remains challenging. While physics-inspired dynamical sampling has been extensively explored for the Ising case ($q=2$) in the form of Ising machines, its generalization to general $q$-state Potts models remains largely unexplored. To fill this gap, we propose a class of physics-inspired dynamical samplers that directly target general $q$-state Potts models, which we refer to as the oscillator Potts machine (OPM). We show, through theoretical analysis and numerical experiments, that the OPM exhibits a systematic low-energy bias with respect to the underlying Potts energy landscape. Furthermore, we demonstrate, via phase perturbation analysis, that the OPM, as overdamped Langevin dynamics, can be realized with a network of self-sustaining oscillators, demonstrating that the OPM is naturally realizable in hardware using standard technology such as CMOS. We design a small-scale ring-oscillator circuit that implements a three-state OPM and validate its operation through transistor-level simulation. Leveraging the low-energy bias of the OPM for Potts models, we then apply it to large-scale max-$K$-cut problems by mapping these instances to $q$-state Potts Hamiltonians and compare its performance against established algorithms. Our results position the OPM as a promising, physically grounded dynamical system framework for multi-state sampling and combinatorial optimization.

Figures (9)

• Figure 1: Overview of the OPM: (A) The energy function of an OPM associated with a $q$-state Potts Hamiltonian is an equilibrium-preserving relaxation that enables overdamped Langevin dynamics (gradient flow with noise) over continuous phases. Equilibrium-preserving relaxation means the continuous energy $U(\theta)$ is constructed so that every discrete Potts configuration corresponds to a structurally stable equilibrium ("sampling point") of the continuous dynamics, and $U(\theta)$ evaluated at those points reproduces the Potts Hamiltonian (up to constants/rescaling). (B) Continuous phases are mapped back to discrete Potts states via a quantization window, yielding a discrete sampling distribution that remains biased toward low-energy configurations. (C) The dynamics can be realized with a coupled oscillator network using $q$th-harmonic injection locking to produce multi-stable phase states compatible with hardware implementation.
• Figure 2: Two different relaxations of a $q$-state Potts Hamiltonian: the non-equilibrium-preserving relaxation (upper) and the equilibrium-preserving relaxation (lower).
• Figure 3: A schematic diagram of the circuit design of the OPM. Example with a three-stage ring oscillator (applicable to any odd-stage ring oscillator). There are $q-1$ band-pass filters with center frequencies $f,2f,\ldots,(q-1)f$ and $q-1$ tunable delay components for each oscillator.
• Figure 4: Topology of the oscillators.
• Figure 5: The evolution of waveforms of four ring oscillators without a $3$rd harmonic injection. The formation time of the three-state phase pattern is about $2{\rm \mu s}$, shown in the label 2.