We examine quantum heuristics that utilize Mixer Hamiltonians, which facilitate the restriction of the search space to a specific subspace and support the implementation of warm-start strategies for solving the Traveling Salesman Problem (TSP). These approaches, based on Mixer Hamiltonians, can be integrated into the Quantum Approximate Optimization Algorithm (QAOA) [1], with the Mixer serving as a mapping function that transforms qubit strings into feasible solution sets. Initially, we introduce a swap-based mixer specifically designed for the TSP, ensuring that only qubit strings corresponding to valid TSP solutions are explored during the QAOA process. Subsequently, we present a warm-start technique that initializes QAOA with a solution derived from any classical heuristic, thereby facilitating faster convergence.