Compressed Quadratization of Higher Order Binary Optimization Problems

TL;DR

This paper introduces a novel method for reducing the degree of higher-order binary optimization problems directly in Ising space, resulting in more compact QUBO representations suitable for resource-limited quantum hardware.

Contribution

It proves the necessity of two variables for degree reduction in Ising space and presents a direct reduction method that improves resource efficiency.

Findings

Degree reduction in Ising space requires two variables per term.

The proposed method produces more compact QUBO formulations for sparse problems.

Direct Ising space reduction avoids the need for space conversion, saving resources.

Abstract

Recent hardware advances in quantum and quantum-inspired annealers promise substantial speedup for solving NP-hard combinatorial optimization problems compared to general-purpose computers. These special-purpose hardware are built for solving hard instances of Quadratic Unconstrained Binary Optimization (QUBO) problems. In terms of number of variables and precision of these hardware are usually resource-constrained and they work either in Ising space {-1,1} or in Boolean space {0,1}. Many naturally occurring problem instances are higher-order in nature. The known method to reduce the degree of a higher-order optimization problem uses Rosenberg's polynomial. The method works in Boolean space by reducing the degree of one term by introducing one extra variable. In this work, we prove that in Ising space the degree reduction of one term requires the introduction of two variables. Our proposed method of degree reduction works directly in Ising space, as opposed to converting an Ising polynomial to Boolean space and applying previously known Rosenberg's polynomial. For sparse higher-order Ising problems, this results in a more compact representation of the resultant QUBO problem, which is crucial for utilizing resource-constrained QUBO solvers.