# Analysis of the Relation between Quadratic Unconstrained Binary Optimization (QUBO) and the Spin Glass Ground-State Problem

## TL;DR

This paper explores the relationship between QUBO and spin glass problems, revealing that large external fields in test instances lead to redundancies and simplified solutions, impacting solver performance and heuristic development.

## Contribution

It provides new insights into the structure of QUBO instances, demonstrates the effects of external fields, and introduces a novel heuristic adaptation and evaluation method.

## Key findings

- Large external fields cause spins to align with fields, reducing problem complexity.
- QUBO test instances contain redundancies that affect solver performance.
- The Extremal Optimization heuristic was adapted for QUBO and evaluated.

## Abstract

We analyze the transformation of QUBO from its conventional Boolean presentation into an equivalent spin glass problem with coupled $\pm1$ spin variables exposed to a site-dependent external field. We find that in a widely used testbed for QUBO these fields tend to be rather large compared to the typical coupling and many spins in each optimal configurations simply align with the fields irrespective of their constraints. Thereby, the testbed instances tend to exhibit large redundancies - seemingly independent variables which contribute little to the hardness of the problem, however. We demonstrate various consequences of this insight, for QUBO solvers as well as for heuristics developed for finding spin glass ground states. To this end, we implement the Extremal Optimization (EO) heuristic, in a new adaptation for the QUBO problem. We also propose a novel way to assess the quality of heuristics for increasing problem sizes based on asymptotic scaling.