Solving larger Travelling Salesman Problem networks with a penalty-free Variational Quantum Algorithm

The Travelling Salesman Problem (TSP) is a well-known NP-Hard combinatorial optimisation problem, with industrial use cases such as last-mile delivery. Although TSP has been studied extensively on quantum computers, it is rare to find quantum solutions of TSP network with more than a dozen locations. In this paper, we present high quality solutions in noise-free Qiskit simulations of networks with up to twelve locations using a hybrid penalty-free, circuit-model, Variational Quantum Algorithm (VQA). Noisy qubits are also simulated. To our knowledge, this is the first successful VQA simulation of a twelve-location TSP on circuit-model devices. Multiple encoding strategies, including factorial, non-factorial, and Gray encoding are evaluated. Our formulation scales as mathcal{O}\(nlog₂(n\)) qubits, requiring only 29 qubits for twelve locations, compared with over 100 qubits for conventional approaches scaling as mathcal{O}\(n²\). Computational time is further reduced by almost two orders of magnitude through the use of Simultaneous Perturbation Stochastic Approximation (SPSA) gradient estimation and cost-function caching. We also introduce a novel machine-learning model, and benchmark both quantum and classical approaches against a Monte Carlo baseline. The VQA outperforms the classical machine-learning approach, and performs similarly to Monte Carlo for the small networks simulated. Additionally, the results indicate a trend toward improved performance with problem size, outlining a pathway to solving larger TSP instances on quantum devices.