The Traveling Salesman Problem (TSP) is a canonical example of an NP-hard combinatorial optimization problem that has extensive applications in the context of logistics and routing and network design, where classical methods cannot be used in practice due to the large scale of the instances. The approach includes extensive research on a hybrid quantum and classical optimisation model, termed Grover-ACO, that combines the Grover quantum amplitude amplification with the elitist selection stage of the Ant Colony Optimization (ACO). This study can be described as an algorithm that includes candidate pool formation, quantum oracle design, amplitude amplification, and pheromone update algorithms, and give a mathematical model of the hybrid algorithm. The evaluation of standard TSPLIB benchmark instances is conducted experimentally on a quantum circuit simulator, and performance is measured in terms of the best and average tour costs, as well as computational overhead. Findings indicate that GroverACO is consistently able to achieve better quality solutions and discourage premature convergence than classical ACO, especially on medium and large-scale problems, albeit at a higher computational budget due to quantum simulation.
Reddy et al. (Mon,) studied this question.