Scalable Quantum Walk-Based Heuristics for the Minimum Vertex Cover Problem
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913158619201536 |
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| author | Luiz, F. S. Iwakami, A. K. F. Moraes, D. H. de Oliveira, M. C. |
| author_facet | Luiz, F. S. Iwakami, A. K. F. Moraes, D. H. de Oliveira, M. C. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scalable Quantum Walk-Based Heuristics for the Minimum Vertex Cover Problem Luiz, F. S. Iwakami, A. K. F. Moraes, D. H. de Oliveira, M. C. Quantum Physics 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. |
| title | Scalable Quantum Walk-Based Heuristics for the Minimum Vertex Cover Problem |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2512.02940 |