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

TL;DR

This paper introduces a scalable quantum heuristic for the Minimum Vertex Cover problem using continuous-time quantum walks, which outperforms classical heuristics in various graph types and requires fewer quantum resources.

Contribution

It presents a novel quantum algorithm leveraging CTQWs with a dynamic decoupling mechanism and efficient encoding, advancing quantum heuristics for combinatorial optimization.

Findings

Outperforms classical heuristics in approximation quality.

Requires only logarithmic qubits, reducing quantum resource needs.

Demonstrates robustness across different network topologies.

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 Erd\H{o}s--R\'enyi, Barab\'asi--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.