Hybrid Quantum-Classical Optimisation of Traveling Salesperson Problem

TL;DR

The paper addresses the NP-hard Traveling Salesperson Problem (TSP) by proposing a hybrid quantum-classical workflow implemented in IBM Qiskit Runtime to exploit quantum exploration with classical refinement. It encodes TSP as a QUBO, uses QAOA-inspired circuits on a 127-qubit device, and augments the process with K-Means clustering for problem decomposition and Random Forest noise mitigation to stabilise solutions. Empirical results for 4–8 cities show that quantum-only approaches are worse than classical baselines (up to 21.7%), while the hybrid quantum-classical approach with ML reduces this gap (down to 11.3% for 8 cities) but remains suboptimal relative to classical solvers. The findings highlight the potential of quantum-enhanced methods in combinatorial optimisation, demonstrating improved robustness and scalability with ML-assisted noise handling, while underscoring the need for hardware and algorithmic advances to achieve quantum advantage.

Abstract

The Traveling Salesperson Problem (TSP) is a fundamental NP-hard optimisation challenge with widespread applications in logistics, operations research, and network design. While classical algorithms effectively solve small to medium-sized instances, they struggle with scalability due to exponential complexity. In this work, we present a hybrid quantum-classical approach that leverages IBM's Qiskit Runtime to integrate quantum optimisation techniques with classical machine learning methods, specifically K-Means clustering and Random Forest classifiers. These machine learning components aid in problem decomposition and noise mitigation, improving the quality of quantum solutions. Experimental results for TSP instances ranging from 4 to 8 cities reveal that the quantum-only approach produces solutions up to 21.7% worse than the classical baseline, while the hybrid method reduces this cost increase to 11.3% for 8 cities. This demonstrates that hybrid approaches improve solution quality compared to purely quantum methods but remain suboptimal compared to classical solvers. Despite current hardware limitations, these results highlight the potential of quantum-enhanced methods for combinatorial optimisation, paving the way for future advancements in scalable quantum computing frameworks.

Figures (5)

• Figure 1: Graph representation of a TSP instance. Cities are depicted as nodes, while weighted edges represent pairwise travel distances. The objective is to determine the shortest possible route that visits each city exactly once before returning to the starting point. The fully connected structure reflects the assumption that direct travel is possible between any two cities.
• Figure 2: Traveling Salesperson Problem Optimisation - Hybrid Quantum-Classical Method Architecture. An overview of the hybrid quantum-classical workflow architecture used to solve the TSP. The diagram highlights the integration of quantum and classical components, showing how quantum circuits are used to explore solutions while classical algorithms refine and optimise them iteratively. The workflow includes transpilation, parameter optimisation, and noise mitigation.
• Figure 3: Visualization of K-Means clustering applied to the TSP. Cities are grouped into distinct clusters, represented with unique colours and styled boundaries. Red crosses mark the centroids of each cluster. This preprocessing step reduces the problem complexity, enabling efficient utilisation of quantum and classical resources for smaller subproblems.
• Figure 4: Solution cost comparison for classical, quantum, and hybrid quantum-classical methods for TSP instances of varying sizes (4–8 cities). The figure includes confidence intervals to show variability and highlights the competitive performance of hybrid approaches, especially for larger problem sizes.
• Figure 5: A heat map illustrating the relative performance deviation of hybrid quantum-classical approaches over classical methods. The data highlights cost efficiency and runtime benefits for larger TSP instances, showcasing the scalability and potential of hybrid workflows enhanced by machine learning.