Scalable Optimization with GIS-PIM: A Generalized Integer-State Probabilistic Ising Machine

Physics-inspired hardware solvers based on Ising machines have gained significant attention for addressing combinatorial optimization problems across diverse domains, including logistics, communication networks, and financial optimization. While the problems in these domains typically involve integer states, they are often reformulated to fit the quadratic unconstrained binary optimization (QUBO) framework supported by Ising machines. This is usually achieved by introducing penalty terms, which can degrade solution quality. In this work, we present a generalized integer-state probabilistic Ising machine (GIS-PIM) that natively supports integer-state encoding using compact binary representations. By avoiding one-hot encoding and associated constraints, GIS-PIM efficiently explores the solution space with fewer spin variables. The framework is implemented on a GPU and evaluated on publicly available graph coloring benchmarks, including both synthetic and realworld datasets. It demonstrates strong scalability, solving problem instances with up to 13 colors and graph sizes reaching approximately 20,000 nodes, a level not previously reported for other Ising-based methods. Our results indicate that the proposed approach: i) achieves solution accuracies exceeding 98%, competitive with leading classical methods such as Tabucol and graph neural network-based solver; and ii) significantly performs better over state-of-the-art simulated bifurcation (SBM) and QUBObased probabilistic Ising machines (Qu-PIM), achieving higher success probability and faster time-to-solution.