Generalized Grover search algorithm for traveling salesman problem (TSP)

Abstract

The search problem is to find the solution, in a given set, that satisfies certain conditions. Grover algorithm is probabilistic such that quantum states corresponding to solutions are observed in high probabilities and it is faster than any classical ones. Traveling salesman problem (TSP) is to find a minimal-cost tour for given cities. Observing the similarities between the two problems, in this work, we generalize Grover algorithm and attempt to design a quantum algorithm to solve TSP. For the purpose, we introduce rather generalized oracle which marks arbitrary phases on all the quantum states as well as the solutions. The generalized oracle, called a cost oracle, encodes the cost of tours to the arbitrary phases. We show that the probability of getting the quantum state corresponding to the minimal-cost tour can be maximized by adjusting another phase operation.