Systematic and Efficient Construction of Quadratic Unconstrained Binary Optimization Forms for High-order and Dense Interactions

TL;DR

This paper introduces a novel quadratization method for transforming complex, high-order, dense interactions in ML-related problems into QUBO forms, enabling efficient quantum annealing solutions.

Contribution

It presents a new quadratization approach using rectified linear unit bases that can handle complex nonlinear and dense interactions in ML problems, expanding QA applicability.

Findings

Numerical and analytical validation of the quadratization method

Successful integration of quadratization with QA for black-box optimization

Demonstrated potential for applying QA to complex ML problems

Abstract

Quantum Annealing (QA) can efficiently solve combinatorial optimization problems whose objective functions are represented by Quadratic Unconstrained Binary Optimization (QUBO) formulations. For broader applicability of QA, quadratization methods are used to transform higher-order problems into QUBOs. However, quadratization methods for complex problems involving Machine Learning (ML) remain largely unknown. In these problems, strong nonlinearity and dense interactions prevent conventional methods from being applied. Therefore, we model target functions by the sum of rectified linear unit bases, which not only have the ability of universal approximation, but also have an equivalent quadratic-polynomial representation. In this study, the proof of concept is verified both numerically and analytically. In addition, by combining QA with the proposed quadratization, we design a new black-box optimization scheme, in which ML surrogate regressors are inputted to QA after the quadratization process.