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IEEE International Conference on Quantum Control, Computing and Learning (qCCL)

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

Siddharth Chander

Published Jun 25, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1109/qccl65142.2025.11159046

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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.
The Quantum Approximate Optimization Algorithm (QAOA) is an algorithmic framework for finding approximate solutions to combinatorial optimization problems. It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians. To fit this framework, one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians. In this paper, for the well-known NP-hard Traveling Salesman Problem (TSP), we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian. Moreover, we map edges (routes) connecting each pair of cities to qubits, which decreases the search space significantly in comparison to other approaches. As a result, our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach. We argue the formalization approach presented in this paper would lead to a generalized framework for finding, in the context of QAOA, high-quality approximate solutions to NP optimization problems.
This thesis explores quantum computing for combinatorial optimization through the Traveling Salesman Problem (TSP), which aims to find a minimum-cost Hamiltonian cycle visiting each city exactly once. Using Qiskit, we implement the Quantum Approximate Optimization Algorithm (QAOA) on both simulators and quantum hardware, and compare its performance with that of classical exact optimization via the Gurobi solver. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with degree and subtour constraints embedded using quadratic penalties. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound solvers outperform it in both solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, the optimality gap grows with problem size, reflecting sensitivity to variational parameter tuning, penalty scaling, and hardware noise. Among classical optimizers for QAOA, COBYLA offers the most effective tradeoff between efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.
The traveling salesman problem (TSP) is one of the most often-used NP-hard problems in computer science to study the effectiveness of computing models and hardware platforms. In this regard, it is also heavily used as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this paper, we tackle the TSP using the quantum approximate optimization algorithm (QAOA) approach by formulating it as an optimization problem. By adopting an improved qubit encoding strategy and a layer-wise learning optimization protocol, we present numerical results obtained from the gate-based digital quantum simulator, specifically targeting TSP instances with 3, 4, and 5 cities. We focus on the evaluations of three distinctive QAOA mixer designs, considering their performances in terms of numerical accuracy and optimization cost. Notably, we find that a well-balanced QAOA mixer design exhibits more promising potential for gate-based simulators and realistic quantum devices in the long run, an observation further supported by our noise model simulations. Furthermore, we investigate the sensitivity of the simulations to the TSP graph. Overall, our simulation results show that the digital quantum simulation of problem-inspired ansatz is a successful candidate for finding optimal TSP solutions.
The Multiple Traveling Salesman Problem (m-TSP) is a well-known combinatorial optimisation problem with diverse applications in various fields, including transportation, logistics, and network design. The m-TSP involves finding the optimal routes for multiple salesmen to visit a set of cities, minimising the total travel distance while ensuring that each city is visited exactly once. Traditional computational approaches struggle to efficiently solve large-scale instances of the m-TSP due to its inherent complexity. In recent years, quantum computing has emerged as a promising field for tackling combinatorial optimisation problems. In this study, we investigate the application of the Quantum Approximate Optimisation Algorithm (QAOA) to solve the m-TSP, leveraging the inherent parallelism and quantum computational power to potentially obtain high-quality solutions.
The Traveling Salesman Problem (TSP) is a fundamental challenge in combinatorial optimization, widely applied in logistics and transportation. As the size of TSP instances grows, traditional algorithms often struggle to produce high-quality solutions within reasonable timeframes. This study investigates the potential of the Quantum Approximate Optimization Algorithm (QAOA), a hybrid quantum-classical method, to solve TSP under realistic constraints. We adopt a QUBO-based formulation of TSP that integrates real-world logistical constraints reflecting operational conditions, such as vehicle capacity, road accessibility, and time windows, while ensuring compatibility with the limitations of current quantum hardware. Our experiments are conducted in a simulated environment using high-performance computing (HPC) resources to assess QAOA's performance across different problem sizes and quantum circuit depths. In order to improve scalability, we propose clustering QAOA (Cl-QAOA), a hybrid approach combining classical machine learning with QAOA. This method decomposes large TSP instances into smaller sub-problems, making quantum optimization feasible even on devices with a limited number of qubits. The results offer a comprehensive evaluation of QAOA's strengths and limitations in solving constrained TSP scenarios. This study advances quantum optimization and lays groundwork for future large-scale applications.
Many relevant problems in industrial settings result in NP-hard optimization problems, such as the Capacitated Vehicle Routing Problem (CVRP) or its reduced variant, the Travelling Salesperson Problem (TSP). Even with today's most powerful classical algorithms, the CVRP is challenging to solve classically. Quantum computing may offer a way to improve the time to solution, although the question remains open as to whether Noisy Intermediate-Scale Quantum (NISQ) devices can achieve a practical advantage compared to classical heuristics. The most prominent algorithms proposed to solve combinatorial optimization problems in the NISQ era are the Quantum Approximate Optimization Algorithm (QAOA) and the more general Variational Quantum Eigensolver (VQE). However, implementing them in a way that reliably provides high-quality solutions is challenging, even for toy examples. In this work, we discuss decomposition and formulation aspects of the CVRP and propose an application-driven way to measure solution quality. Considering current hardware constraints, we reduce the CVRP to a clustering phase and a set of TSPs. For the TSP, we extensively test both QAOA and VQE and investigate the influence of various hyperparameters, such as the classical optimizer choice and strength of constraint penalization. Results of QAOA are generally of limited quality because the algorithm does not reach the energy threshold for feasible TSP solutions, even when considering various extensions such as recursive and constraint-preserving mixer QAOA. On the other hand, the VQE reaches the energy threshold and shows a better performance. Our work outlines the obstacles to quantum-assisted solutions for real-world optimization problems and proposes perspectives on how to overcome them.