Mathematical Mechanism on Dynamical System Algorithms of the Ising Model

TL;DR

This paper uncovers the mathematical principles behind dynamical system algorithms for the Ising model using Morse theory and variational methods, providing a unified understanding of several existing algorithms and their convergence properties.

Contribution

It introduces a mathematical framework explaining how dynamical system algorithms minimize a continuous function to solve Ising problems, linking physical algorithms to rigorous mathematical principles.

Findings

Dynamical algorithms minimize a continuous function with local minima corresponding to Ising solutions.

The global minimum of the continuous function yields the Ising model's optimal solution.

Analysis of convergence properties via transit and capture in celestial mechanics context.

Abstract

Various combinatorial optimization NP-hard problems can be reduced to finding the minimizer of an Ising model, which is a discrete mathematical model. It is an intellectual challenge to develop some mathematical tools or algorithms for solving the Ising model. Over the past decades, some continuous approaches or algorithms have been proposed from physical, mathematical or computational views for optimizing the Ising model such as quantum annealing, the coherent Ising machine, simulated annealing, adiabatic Hamiltonian systems, etc.. However, the mathematical principle of these algorithms is far from being understood. In this paper, we reveal the mathematical mechanism of dynamical system algorithms for the Ising model by Morse theory and variational methods. We prove that the dynamical system algorithms can be designed to minimize a continuous function whose local minimum points give all the candidates of the Ising model and the global minimum gives the minimizer of Ising problem. Using this mathematical mechanism, we can easily understand several dynamical system algorithms of the Ising model such as the coherent Ising machine, the Kerr-nonlinear parametric oscillators and the simulated bifurcation algorithm. Furthermore, motivated by the works of C. Conley, we study transit and capture properties of the simulated bifurcation algorithm to explain its convergence by the low energy transit and capture in celestial mechanics. A detailed discussion on $2$-spin and $3$-spin Ising models is presented as application.