The Travelling Salesman Problem (TSP) constitutes a fundamental Combinatorial Optimisation Problem (COP) with wide-ranging applications in logistics, transportation design and production scheduling. However, the best-known exact classical algorithms exhibit exponential time complexity, rendering large instances intractable in practice and motivating the exploration of alternative paradigms such as quantum computing. In this work, we implement the Quadratic Unconstrained Binary Optimisation (QUBO) formulation and its generalisation, Tensor Quadratic Unconstrained D-ary Optimisation (T-QUDO), which extends quantum circuit encoding from qubits to qudits, for a TSP variant. This variant seeks to determine a minimum-cost route while incorporating a timedependent cost function and precedence constraints, and remains largely unexplored with T-QUDO despite its relevance to dynamic travel planning. The formulations are encoded within the Quantum Approximate Optimisation Algorithm (QAOA) framework and analysed under noiseless simulations with the Cirq and CUDA-Q libraries, aiming to assess the potential advantages of qudit-based representations over qubit-based ones, particularly in terms of expressivity, resource efficiency, and solution quality.
The Travelling Salesman Problem (TSP) constitutes a fundamental Combinatorial Optimisation Problem (COP) with wide-ranging applications in logistics, transportation design and production scheduling. However, the best-known exact classical algorithms exhibit exponential time complexity, rendering large instances intractable in practice and motivating the exploration of alternative paradigms such as quantum computing. In this work, we implement the Quadratic Unconstrained Binary Optimisation (QUBO) formulation and its generalisation, Tensor Quadratic Unconstrained D-ary Optimisation (T-QUDO), which extends quantum circuit encoding from qubits to qudits, for a TSP variant. This variant seeks to determine a minimum-cost route while incorporating a timedependent cost function and precedence constraints, and remains largely unexplored with T-QUDO despite its relevance to dynamic travel planning. The formulations are encoded within the Quantum Approximate Optimisation Algorithm (QAOA) framework and analysed under noiseless simulations with the Cirq and CUDA-Q libraries, aiming to assess the potential advantages of qudit-based representations over qubit-based ones, particularly in terms of expressivity, resource efficiency, and solution quality.
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.
Abstract The traveling salesman problem (TSP) forms the foundation for many logistics and optimization tasks. At the same time, this problem is NP-hard and requires significant computational resources. Quantum computing promises to substantially accelerate computations or increase solution accuracy. The TSP can be conveniently mapped onto the quadratic unconstrained binary optimization (QUBO) formulation, which in turn can be solved using variational quantum eigensolver (VQE) and quantum approximate optimization algorithm (QAOA) methods, but this mapping requires a quadratic number of qubits. The higher-order binary optimization (HOBO) formulation requires significantly fewer qubits, but it contains higher-order terms. In our work, we thoroughly investigate both formulations and provide a practical recipe for solving the TSP. We give a comprehensive study and compare in detail QUBO and HOBO formulations.
This study explores quantum computing for combinatorial optimization, focusing on the Traveling Salesman Problem (TSP). Using Qiskit, the Quantum Approximate Optimization Algorithm (QAOA) is implemented on simulators and quantum hardware, and compared with classical optimization using Gurobi. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with quadratic penalties enforcing degree and subtour constraints. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound methods achieve better solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, performance degrades with problem size due to sensitivity to parameter tuning, penalty scaling, and hardware noise. Among the tested classical optimizers, COBYLA provides the best balance of 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 a cornerstone of combinatorial optimization with widespread applications in logistics and transportation. As problem sizes increase, classical algorithms often fail to deliver high-quality solutions within practical time constraints. This paper explores the use of the Quantum Approximate Optimization Algorithm (QAOA), a hybrid quantum-classical algorithm, to address TSP instances under realistic conditions. We present a QUBO-based formulation of the TSP that integrates practical constraints reflecting real-world conditions—such as vehicle capacity, road accessibility, and time windows—while maintaining compatibility with limitations of current quantum hardware. Our analysis is conducted in a simulated environment, leveraging high-performance computing (HPC) resources to evaluate the algorithm’s performance across varying problem sizes and circuit depths. This approach enables a comprehensive assessment of QAOA’s capabilities and limitations in solving constrained TSP scenarios, thereby laying the groundwork for its deployment on future large-scale quantum hardware.
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.
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.