Non-binary dynamical Ising machines for combinatorial optimization

TL;DR

This work challenges the standard requirement that dynamical Ising machines converge to binary states by introducing the V2 non-binary model, which represents relaxed spins as a binary component $\boldsymbol{\sigma}$ and a continuous remainder $\mathbf{X}$ with $\xi_i = \sigma_i + X_i + 4k_i$. The authors define a relaxed cut $C_R$ that evolves under continuous dynamics to a steady state where the discrete cut $C(\boldsymbol{\sigma})$ is read from the binary part, eliminating the need for external rounding in many cases. They demonstrate how graph coloring, Latin squares, and Sudoku can be formulated as max-cut problems on augmented graphs and solved via the V2 dynamics, with stable definite color states emerging even when the terminal state is non-binary. The results indicate potential for scalable electronic accelerators of combinatorial optimization by leveraging continuous-time dynamics that inherently encode feasible discrete solutions. This non-binary approach broadens the design space for Ising machines and highlights robust solution-feasibility without heavy post-processing.

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.

Figures (11)

• Figure 1: Examples of cut-counting functions: $\Phi_{\mathrm{I}}$ is the discrete function of the binary Ising model, $\Phi_{R}(\boldsymbol{\xi})$ are relaxed functions with $M = \mathrm{SDP}, \mathrm{Tr}, \mathrm{V}_2$ corresponding to rank-$2$ SDP relaxation Burer2002, the triangular model from Refs. Shukla2024ScalableMachinesShukla2023CustomHeuristic, and the $\mathrm{V}_2$ model considered in the present paper. For the relaxed cut-counting functions, only one period is shown.
• Figure 2: The phase space of the dynamical variables in the $V_2$ model is a wedge sum of two circles with a circumference of $2$. (a) Range of an state-vector element $(\sigma, X)$, with $X$ depicted as an open interval. (b) $(\sigma, X)$ with $X$ shown as a circle to the visualize the continuity of the transition between states $(\sigma,\pm1)$ and $(-\sigma,\mp1)$.
• Figure 3: The dependence of the probability of convergence to a definite color state for a single node graph on the number of colors. Three machines are shown: the coherent Ising machine (CIM) yamamotoCoherent2017 and based on the triangular Shukla2024ScalableMachines and $V_2$ models.
• Figure 4: An example of the progression of a spin complex representing $K = 7$ colors from an initial generic state to a state with definite color. The stationary red vertical arrow shows the auxiliary spin. The label at the center shows the identified color ($0$ corresponds to the state that does not represent a valid color). Shown time is measured in units of the equations of motion [Eq. \ref{['eq:k-color_eqm']}].
• Figure 5: Example coloring problem. (a) 6-node problem graph; (b) possible coloring of the 6-node graph; (c) Ising model of the graph, the central node in the graph is the auxiliary node with the fixed spin.

Theorems & Definitions (2)

• proof
• proof