Abstract
This paper investigates quantum heuristics based on Mixer Hamiltonians, which allow the search to be restricted to a specific subspace and enable warm-start strategies for solving the Traveling Salesman Problem (TSP). Approaches involving Mixer Hamiltonians can be integrated into the Quantum Approximate Optimization Algorithm (QAOA), where the Mixer acts as a mapping function that transforms qubit strings into feasible solution sets. We first introduce a swap-based mixer tailored to the TSP, which ensures that only qubit strings representing valid TSP solutions are explored during the QAOA process. Second, we propose a warm-start technique that initializes QAOA with a solution generated by any classical heuristic, thereby promoting faster convergence. These two contributions are combined into a Warm-Start QAOA framework with a Swap-Based Mixer, leveraging both structural and initialization advantages. Experimental results on a custom TSP instance involving five customers demonstrate the effectiveness of this approach, providing, for the first time, a viable integration of warm-start and swap-based mixers for the TSP within a quantum optimization framework.