Mean Field Approximation for solving QUBO problems

TL;DR

This paper demonstrates that mean field annealing, inspired by statistical physics and quantum mechanics, effectively approximates solutions to QUBO problems, achieving near-optimal results on benchmark graphs.

Contribution

It shows the equivalence of statistical physics and quantum mechanical approaches in mean field annealing for QUBO problems and provides a simple gradient-based method for large instances.

Findings

Achieved best-known cut values on many graphs.

Proposed a simple, gradient-based minimization approach.

Validated the method on the Maximum Cut problem with G-sets.

Abstract

The Quadratic Unconstrained Binary Optimization (QUBO) problems are NP hard; thus, so far, there are no algorithms to solve them efficiently. There are exact methods like the Branch-and-Bound algorithm for smaller problems, and for larger ones, many good approximations like stochastic simulated annealing for discrete variables or the mean field annealing for continuous variables. This paper will show that the statistical physics approach and the quantum mechanical approach in the mean field annealing give the same result. We examined the Ising problem, which is an alternative formulation of the QUBO problem. Our methods consist of a set of simple gradient-based minimizations with continuous variables, thus easy to simulate. We benchmarked our methods with solving the Maximum Cut problem with the G-sets. In many graphs, we could achieve the best-known Cut Value.