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arXiv:2512.02940

Scalable Quantum Walk-Based Heuristics for the Minimum Vertex Cover Problem

F. S. Luiz, A. K. F. Iwakami, D. H. Moraes, M. C. de Oliveira

quant-phSubmitted Dec 2, 2025• Announced Dec 3, 2025• 0 citations• Revised May 23, 2026 (v2)
v1 · Dec 2, 2025v2 · May 23, 2026
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Abstract

We propose a novel heuristic quantum algorithm for the Minimum Vertex Cover (MVC) problem based on continuous-time quantum walks (CTQWs). In this framework, the coherent propagation of a quantum walker over a graph encodes its structural properties into state amplitudes, enabling the identification of highly influential vertices through their transition probabilities. To enhance stability and solution quality, we introduce a dynamic decoupling (``freezing'') mechanism that isolates vertices already selected for the cover, preventing their interference in subsequent iterations of the algorithm. The method employs a compact binary encoding, requiring only $\lceil \log_2 (V)\rceil$ qubits to represent a graph with $V$ vertices, resulting in an exponential reduction of quantum resources compared to conventional vertex-based encodings. We benchmark the proposed heuristic against exact solutions obtained via Mixed-Integer Linear Programming (MILP) and against established classical heuristics, including Simulated Annealing, FastVC, and the 2-Approximation algorithm, across Erdős--Rényi, Barabási--Albert and regular random graph ensembles. Our results demonstrate that the CTQW-based heuristic consistently achieves superior approximation ratios and exhibits remarkable robustness with respect to network topology, outperforming classical approaches in both heterogeneous and homogeneous structures. These findings indicate that continuous-time quantum walks, when combined with topology-independent decoupling strategies, provide a powerful paradigm for large-scale combinatorial optimization and complex network control, with potential applications spanning infrastructure resilience, epidemic containment, sensor network optimization, and biological systems analysis.

Comments
14 pages, 8 figures

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Link prediction is a fundamental task in network science that involves estimating the likelihood of missing or future connections between nodes based on the structural patterns of a graph. While Graph Neural Networks (GNNs) have become a dominant approach, Random Walk (RW)-based methods remain highly relevant due to their lower computational demands in large-scale networks. Although recent hybrid quantum-classical walks have shown promise in exploring graph structures, existing continuous-time models are computationally expensive as they require solving high-dimensional systems of differential equations. In this paper, we propose a discrete version of a recent hybrid quantum-classical random walk algorithm that significantly improves computational efficiency by utilizing state vector representations instead of density matrices. Our method, inspired by the Quantum Jumps approach, alternates between coherent evolution and measurement-driven state collapse to generate trajectories for node embeddings. We integrate this algorithm into a link prediction pipeline and evaluate its performance across various synthetic topologies, including Grid, Power-Law, Erdös-Rényi, Barabási-Albert, and Community graphs. The experimental results demonstrate that the proposed discrete hybrid walks consistently outperform classical random walk approaches in Community and Grid topologies, showing that quantum-inspired dynamics can capture complex structural information more effectively than classical dynamics for specific topologies.