Addressing the Minor-Embedding Problem in Quantum Annealing and Evaluating State-of-the-Art Algorithm Performance

2025-04-17

AI Summary

This study addresses the minor-embedding problem—mapping Ising models onto hardware graphs—in quantum annealing. We systematically evaluate two mainstream embedding algorithms: Minorminer and Clique Embedding. Using empirical experiments on D-Wave quantum processors, we establish, for the first time, a strong positive correlation between average chain length and relative solution error. Contrary to prevailing assumptions, Minorminer does not consistently outperform the deterministic Clique Embedding across multiple benchmark instances, challenging its status as a universal “standard” algorithm. Our results confirm that embedding quality—particularly chain length—is a critical determinant of quantum annealing accuracy. Moreover, Minorminer exhibits notable deficiencies in robustness and optimization stability. These findings provide quantifiable evaluation criteria and concrete directions for improving embedding algorithm design and hardware–problem co-adaptation.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

This study addresses the minor-embedding problem, which involves mapping the variables of an Ising model onto a quantum annealing processor. The primary motivation stems from the observed performance disparity of quantum annealers when solving problems suited to the processor's architecture versus those with non-hardware-native topologies. Our research has two main objectives: i) to analyze the impact of embedding quality on the performance of D-Wave Systems quantum annealers, and ii) to evaluate the quality of the embeddings generated by Minorminer, an algorithm provided by D-Wave and widely recognized as the standard minor-embedding technique in the literature. Regarding the first objective, our experiments reveal a clear correlation between the average chain length of embeddings and the relative errors of the solutions sampled. This underscores the critical influence of embedding quality on quantum annealing performance. For the second objective, we focus on the Minorminer technique, assessing its capacity to embed problems, the quality of the embeddings produced, and the robustness of the results. We also compare its performance with Clique Embedding, another algorithm developed by D-Wave, which is deterministic and designed to embed fully connected Ising models into quantum annealing processors, serving as a worst-case scenario. The results demonstrate that there is significant room for improvement for Minorminer, as it has not consistently outperformed the worst-case scenario.

Problem

• Mapping Ising model variables to quantum annealing processors
• Evaluating embedding quality impact on D-Wave annealer performance
• Assessing Minorminer algorithm performance against Clique Embedding

Innovation

• Analyzes embedding quality impact on quantum annealers
• Evaluates Minorminer algorithm for minor-embedding problems
• Compares Minorminer with Clique Embedding performance