Performance analysis of the quantum alternating operator ansatz for solving the minimum weighted vertex cover problem

The Quantum Alternating Operator Ansatz (QAOA+) extends the Quantum Approximate Optimization Algorithm (QAOA) to solve constrained combinatorial optimization problems more effectively. In this study, we explore the application of QAOA+ to the Minimum Weighted Vertex Cover (MWVC) problem in graph theory and evaluate its performance through numerical experiments. The results indicate that although QAOA+ can solve the MWVC problem efficiently, the quality of its solution is lower compared to the unweighted Minimum Vertex Cover (MVC) problem on the same graph. The study reveals that vertex weights introduce dependencies between the parameters of different qubit gates in the quantum circuit, which restricts parameter optimization and brings additional complexity to the problem. Other algorithms generally encounter difficulty when moving from solving MVC problems to MWVC problems, with performance degrading on most weighted instances, while classical heuristic algorithms may perform better on weighted instances with the help of empirical rules or heuristic strategies.