A quantum annealing approach to graph node embedding

TL;DR

This work investigates using quantum annealing to learn graph node embeddings by formulating the NE problem as a QUBO suitable for D-Wave hardware. It introduces auxiliary variables to linearize binary products, and several constraint-handling strategies, including penalty methods and augmented Lagrangian approaches (ALM and ALMQ), to produce robust QUBO formulations. The study compares three similarity metrics—Jac with nonzero pairs (Jac), Jac including zero similarities (Jac0), and graph adjacency similarity (adjcy)—and demonstrates that QA-based methods often outperform classical simulated annealing, with ALMQ offering the best overall accuracy across many settings. The findings indicate QA is a viable, scalable alternative for graph-based learning on current quantum hardware, particularly when using adjacency-based similarity, while highlighting hardware limitations and avenues for extending to real-valued embeddings in future work.

Abstract

Node embedding is a key technique for representing graph nodes as vectors while preserving structural and relational properties, which enables machine learning tasks like feature extraction, clustering, and classification. While classical methods such as DeepWalk, node2vec, and graph convolutional networks learn node embeddings by capturing structural and relational patterns in graphs, they often require significant computational resources and struggle with scalability on large graphs. Quantum computing provides a promising alternative for graph-based learning by leveraging quantum effects and introducing novel optimization approaches. Variational quantum circuits and quantum kernel methods have been explored for embedding tasks, but their scalability remains limited due to the constraints of noisy intermediate-scale quantum (NISQ) hardware. In this paper, we investigate quantum annealing (QA) as an alternative approach that mitigates key challenges associated with quantum gate-based models. We propose several formulations of the node embedding problem as a quadratic unconstrained binary optimization (QUBO) instance, making it compatible with current quantum annealers such as those developed by D-Wave. We implement our algorithms on a D-Wave quantum annealer and evaluate their performance on graphs with up to 100 nodes and embedding dimensions of up to 5. Our findings indicate that QA is a viable approach for graph-based learning, providing a scalable and efficient alternative to previous quantum embedding techniques.

Figures (4)

• Figure 1: Comparison of the methods for different classes of input graphs, embedding dimension, and similarity type.
• Figure 2: Error comparison across embedding dimensions for different similarity metrics. Labels indicate the similarity function used in the objective function (first component) and in the error estimation (second component).
• Figure 3: Dependence of QUBO size on the number of nodes, embedding dimension, and similarity metric. The top row displays the number of linear terms (number of variables), while the bottom row shows the number of quadratic terms.
• Figure 4: Scaling behavior of QUBO size and performance metrics for graph adjacency similarity. The first row shows the percentage of embeddable instances, the second row presents the QA error, and the third and fourth rows display the number of linear and quadratic terms, respectively.