A Realizable GAS-based Quantum Algorithm for Traveling Salesman Problem

The paper proposes a quantum algorithm for the traveling salesman problem (TSP) based on the Grover Adaptive Search (GAS), which can be successfully executed on IBM's Qiskit library. Under the GAS framework, there are at least two fundamental difficulties that limit the application of quantum algorithms for combinatorial optimization problems. One difficulty is that the solutions given by the quantum algorithms may not be feasible. The other difficulty is that the number of qubits of current quantum computers is still very limited, and it cannot meet the minimum requirements for the number of qubits required by the algorithm. In response to the above difficulties, we designed and improved the Hamiltonian Cycle Detection (HCD) oracle based on mathematical theorems. It can automatically eliminate infeasible solutions during the execution of the algorithm. On the other hand, we design an anchor register strategy to save the usage of qubits. The strategy fully considers the reversibility requirement of quantum computing, overcoming the difficulty that the used qubits cannot be simply overwritten or released. As a result, we successfully implemented the numerical solution to TSP on IBM's Qiskit. For the seven-node TSP, we only need 31 qubits, and the success rate in obtaining the optimal solution is 86.71%.