Computational Overhead of Locality Reduction in Binary Optimization Problems

TL;DR

This paper investigates how reducing the locality of binary optimization problems to 2-local forms increases their computational difficulty, highlighting the importance of avoiding such reductions in quantum and quantum-inspired solvers.

Contribution

It provides a detailed analysis of the overhead caused by locality reduction and demonstrates its impact on problem hardness using empirical experiments.

Findings

Locality reduction significantly increases problem complexity.

Post-reduction problems are harder to solve with quantum-inspired algorithms.

Avoiding locality reduction can improve the efficiency of solving binary optimization problems.

Abstract

Recently, there has been considerable interest in solving optimization problems by mapping these onto a binary representation, sparked mostly by the use of quantum annealing machines. Such binary representation is reminiscent of a discrete physical two-state system, such as the Ising model. As such, physics-inspired techniques -- commonly used in fundamental physics studies -- are ideally suited to solve optimization problems in a binary format. While binary representations can be often found for paradigmatic optimization problems, these typically result in k-local higher-order unconstrained binary optimization cost functions. In this work, we discuss the effects of locality reduction needed for the majority of the currently available quantum and quantum-inspired solvers that can only accommodate 2-local (quadratic) cost functions. General locality reduction approaches require the introduction of ancillary variables which cause an overhead over the native problem. Using a parallel tempering Monte Carlo solver on Microsoft Azure Quantum, as well as k-local binary problems with planted solutions, we show that post reduction to a corresponding 2-local representation the problems become considerably harder to solve. We further quantify the increase in computational hardness introduced by the reduction algorithm by measuring the variation of number of variables, statistics of the coefficient values, and the population annealing entropic family size. Our results demonstrate the importance of avoiding locality reduction when solving optimization problems.