Non-binary dynamical Ising machines for combinatorial optimization

TL;DR

This paper introduces non-binary dynamical Ising machines that can directly represent discrete solutions in continuous systems, improving scalability and accuracy in solving combinatorial optimization problems like graph coloring and Sudoku.

Contribution

The paper presents a novel class of non-binary dynamical Ising machines that eliminate the need for post-processing by unambiguously encoding discrete states within continuous dynamical systems.

Findings

Successfully applied to graph coloring, Latin squares, and Sudoku.

Demonstrated unambiguous representation of discrete states in continuous systems.

Opened new avenues for scalable electronic optimization accelerators.

Abstract

Dynamical Ising machines achieve accelerated solving of complex combinatorial optimization problems by remapping the convergence to the ground state of the classical spin networks to the evolution of specially constructed continuous dynamical systems. The main adapted principle of constructing such systems is based on requiring that, on the one hand, the system converges to a binary state and, on the other hand, the system's energy in such states mimics the classical Ising Hamiltonian. The emergence of binary-like states is regarded to be an indispensable feature of dynamical Ising machines as it establishes the relation between the machine's continuous terminal state and the inherently discrete solution of a combinatorial optimization problem. This is emphasized by problems where the unknown quantities are represented by spin complexes, for example, the graph coloring problem. In such cases, an imprecise mapping of the continuous states to spin configurations may lead to invalid solutions requiring intensive post-processing. In contrast to such an approach, we show that there exists a class of non-binary dynamical Ising machines without the incongruity between the continuous character of the machine's states and the discreteness of the spin states. We demonstrate this feature by applying such a machine to the problems of finding proper graph coloring, constructing Latin squares, and solving Sudoku puzzles. Thus, we demonstrate that the information characterizing discrete states can be unambiguously presented in essentially continuous dynamical systems. This opens new opportunities in the realization of scalable electronic accelerators of combinatorial optimization.