Abstract
The Minimum Vertex Cover (MVC) problem, a fundamental NP-hard combinatorial optimization task, seeks the smallest set of vertices in a graph such that every edge is incident to at least one selected vertex. I introduce a quantum greedy algorithm for MVC that combines a constraint-preserving quantum ansatz with a classical sequential variable-fixing strategy. Using a single-layer multi-controlled rotation circuit, the quantum state encodes an unbalanced superposition over all feasible covers, enabling greedy selection toward minimal solutions. This hybrid approach preserves feasibility by construction, explores the solution space efficiently, and retains higher-dimensional superpositions throughout the optimization, offering a scalable alternative to penalty-based QAOA for constrained combinatorial optimization on near-term quantum hardware.