This article proposes a new algorithm based on evolutionary computation and quantum computing. It attempts to resolve ordering combinatorial optimization problems, the most well known of which is the traveling salesman problem (TSP). Classic and quantum-inspired genetic algorithms based on binary representations have been previously used to solve combinatorial optimization problems. However, for ordering combinatorial optimization problems, order-based genetic algorithms are more adequate than those with binary representation, since a specialized crossover process can be employed in order to always generate feasible solutions. Traditional order-based genetic algorithms have already been applied to ordering combinatorial optimization problems but few quantum-inspired genetic algorithms have been proposed. The algorithm presented in this paper contributes to the quantum-inspired genetic approach to solve ordering combinatorial optimization problems. The performance of the proposed algorithm is compared with one orderbased genetic algorithm using uniform crossover. In all cases considered, the results obtained by applying the proposed algorithm to the TSP were better, both in terms of processing times and in terms of the quality of the solutions obtained, than those obtained with order-based genetic algorithms.