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

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.