Comparative Analysis of Quantum Approximate Optimization Algorithms for Solving the Travelling Salesperson Problem

The Quantum Approximate Optimization Algorithm (QAOA) was developed to tackle combinatorial optimization problems, such as the Travelling Salesman Problem (TSP), by approximating solutions through quantum mechanics. QAOA utilizes a unitary transformation to prepare a quantum state that encodes a potential solution, with the parameters of the unitary optimized by a classical optimizer. This study evaluates QAOA's performance on the NP-hard TSP, comparing the standard version of the algorithm from Farhi \cite{qaoa} and an enhancement, Warm-Start QAOA \cite{warmstart}, to a classical solver. The TSP seeks the shortest possible route that visits a set of cities, a problem that is amenable to quantum formulation using platforms like Qiskit. Warm-Start QAOA initializes the algorithm's parameters based on a solution to a relaxed version of the problem, offering potential improvements in optimization efficiency. By comparing these quantum algorithms to a classical solver, we provide a comprehensive assessment of their performance across key metrics, including route quality, execution time, and scalability. Through this analysis, we highlight the relative strengths and weaknesses of QAOA and its warm-start variant for solving the TSP, alongside the classical approach, offering insights into their potential for route optimization and practical applications in realworld settings. We conclude that, while the classical solver significantly outperforms QAOA and warm-start QAOA at smaller-scale problem cases, warm-Start QAOA has the potential to yield advantages at a larger scale when run on NISQ-era noisy hardware.