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

TL;DR

This paper demonstrates a scalable, penalty-free Variational Quantum Algorithm for solving larger TSP networks, achieving high-quality solutions with fewer qubits and reduced computational time, including simulations of up to twelve locations.

Contribution

The paper introduces a novel penalty-free VQA approach for TSP, scalable to larger networks, and compares multiple encoding strategies, outperforming classical machine learning methods in simulations.

Findings

Successfully simulated 12-location TSP with quantum algorithms.

Scales as O(n log n) qubits, requiring only 29 qubits for 12 locations.

VQA outperforms classical machine learning and approaches Monte Carlo performance.

Abstract

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 $\mathcal{O}(nlog_2(n))$ qubits, requiring only 29 qubits for twelve locations, compared with over 100 qubits for conventional approaches scaling as $\mathcal{O}(n^2)$. 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.