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

2025-06-10

AI Summary

High-order, strongly nonlinear machine learning optimization problems with dense variable interactions are notoriously difficult to efficiently map onto Quadratic Unconstrained Binary Optimization (QUBO) form—a key bottleneck limiting practical quantum annealing (QA) applications. Method: We propose a provably equivalent quadratic expansion method based on ReLU basis functions, enabling systematic, low-overhead, high-fidelity quadratization. Building upon this, we introduce the first black-box optimization paradigm that tightly integrates machine learning surrogate models (e.g., neural networks) with quantum annealing. Contribution/Results: We provide rigorous theoretical guarantees of equivalence and demonstrate numerically that our approach significantly reduces auxiliary variable count and coupling complexity compared to conventional quadratization methods. Moreover, it enables end-to-end QA-based optimization on representative ML tasks—achieving competitive accuracy, scalability, and hardware compatibility with current quantum annealers.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Convert high-order dense interactions to QUBO forms
• Address quadratization challenges in Machine Learning problems
• Develop black-box optimization combining QA and quadratization

Innovation

• Uses rectified linear unit bases for modeling
• Converts high-order problems into QUBO forms
• Combines QA with ML surrogate regressors