Embedding All Feasible Solutions of Traveling Salesman Problem by Divide-and-Conquer Quantum Search

Quantum search algorithms, such as the Grover algorithm, are investigated as one of the ways to solve a traveling salesman problem (TSP). In TSP, the quantum search starts from an equal superposition of all feasible solutions. The embedding cost of all feasible solutions is up to exponential, i.e., $\mathcal{O}(n!)$ with $n$ cities, and makes it difficult to solve large-scale TSP. Hence, preparing the initial state of TSP at a lower embedding cost is an important issue. Here, we propose a quantum search algorithm based on the divide-and-conquer approach, which divides each constraint term of the TSP into subproblems and applies the quantum search to each subproblem. We investigate the embedding quality of all feasible solutions by comparing our proposed method with the Grover algorithm regarding optimal iterations and success probability. The results show that our quantum divide-and-conquer approach can significantly reduce the embedding cost while the success probability decreases.