Comparative study on the variations of quantum approximate optimization algorithms to the Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is one of the most often-used NP-Hard problems in computer science to study the effectiveness of computing models and hardware platforms. In this regard, it is also being used heavily as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this work, we formulate the symmetric TSP as an optimization problem, which we solve using the quantum approximate optimization algorithm (QAOA) approach. By adopting an improved qubit encoding strategy and a layerwise learning optimization protocol, we obtain numerical results on the gate-based digital quantum simulator for the 3-, 4-, and 5-city TSPs. Specifically, we focus on three QAOA mixer designs to evaluate their performances in terms of numerical accuracy and optimization cost. Based on our results, we propose that a well-balanced QAOA mixer design is more prominent on gate-based simulators or realistic quantum devices in the near future. In addition, we study the sensitivity of TSP graph properties such as graph skewness and penalty weight in the TSP-QAOA simulation. Overall, our results prove digital quantum simulation is a powerful candidate for obtaining the optimal solution to the TSP.