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arXiv:2605.00739

A Resource-Efficient Variational Quantum Framework for the Traveling Salesman Problem

Yuefeng Lin, Chao Zheng, Cong Guo

quant-phSubmitted May 1, 2026• Announced May 4, 2026• 0 citations
arXivHTMLPDF

Abstract

The Traveling Salesman Problem (TSP) is a prototypical combinatorial optimization problem, but its quantum implementation is limited by the O(n^2)-qubit overhead of standard one-hot encodings. Here, we propose a resource-efficient variational quantum framework based on compact binary-register encoding, a permutation-preserving problem-inspired ansatz, and a complementary divide-and-conquer execution strategy. The compact encoding reduces the data-qubit requirement to O(n log n), while the divide-and-conquer formulation lowers the number of qubits required in each local hardware execution to the size of the largest subsystem. Numerical simulations on TSP instances with 4, 5, and 6 cities achieve best average success rates of 100%, 100%, and 95.5%, respectively. A local two-qubit implementation of the divide-and-conquer approximation is further evaluated for a 5-city TSP instance on SpinQ Gemini Pro and SpinQ Triangulum II NMR quantum computers. Taken together, the results indicate how compact encoding and divide-and-conquer execution with classical post-processing can be used to study small combinatorial optimization instances on resource-constrained quantum hardware.

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The Traveling Salesman Problem (TSP) is a widely studied NP-complete optimization challenge with significant theoretical and practical implications. This study proposes a hybrid quantum-classical framework using a Quantum-Inspired Evolutionary Algorithm (QEA) with Sort Gray Binary Encoding to solve the TSP. The proposed method guarantees the generation of valid TSP tours by eliminating invalid solutions. It employs quantum superposition with intrinsic randomness to enhance computational efficiency and scalability. The framework was implemented on cloud-based NISQ platforms, including IBM Quantum and AWS Braket, demonstrating its practicality and effectiveness. Experimental evaluations revealed that the proposed framework successfully solved TSP instances with up to 15 cities, achieving superior performance compared to classical methods and showcasing its ability to scale under NISQ constraints. The results also highlight the potential of hybrid quantum-classical approaches to overcome hardware limitations in current quantum systems. This study establishes a robust hybrid methodology for solving combinatorial optimization problems. It also sets a benchmark for leveraging the capabilities of NISQ-era quantum devices in real-world applications, thereby providing a foundation for future research in hybrid quantum-classical optimization techniques.

IEEE Access•Jan 1, 2025•4 citations
The Traveling Salesman Problem (TSP) is a prototypical combinatorial optimization problem, but its quantum implementation is limited by the O(n^2)-qubit overhead of standard one-hot encodings. Here, we propose a resource-efficient variational quantum framework based on compact binary-register encoding, a permutation-preserving problem-inspired ansatz, and a complementary divide-and-conquer execution strategy. The compact encoding reduces the data-qubit requirement to O(n log n), while the divide-and-conquer formulation lowers the number of qubits required in each local hardware execution to the size of the largest subsystem. Numerical simulations on TSP instances with 4, 5, and 6 cities achieve best average success rates of 100%, 100%, and 95.5%, respectively. A local two-qubit implementation of the divide-and-conquer approximation is further evaluated for a 5-city TSP instance on SpinQ Gemini Pro and SpinQ Triangulum II NMR quantum computers. Taken together, the results indicate how compact encoding and divide-and-conquer execution with classical post-processing can be used to study small combinatorial optimization instances on resource-constrained quantum hardware.
In this paper we present a variational algorithm for the Traveling Salesman Problem (TSP) that combines (i) a compact encoding of permutations, which reduces the qubit requirement too, (ii) an optimize-freeze-reuse strategy: where the circuit topology (``Ansatz'') is first optimized on a training instance by Simulated Annealing (SA), then ``frozen'' and re-used on novel instances, limited to a rapid re-optimization of only the circuit parameters. This pipeline eliminates costly structural research in testing, making the procedure immediately implementable on NISQ hardware. On a set of 404040 randomly generated symmetric instances that span 4−74 - 74−7 cities, the resulting Ansatz achieves an average optimal trip sampling probability of 100%100\%100% for 4 city cases, 90%90\%90% for 5 city cases and 80%80\%80% for 6 city cases. With 7 cities the success rate drops markedly to an average of ∼20%\sim 20\%∼20%, revealing the onset of scalability limitations of the proposed method. The results show robust generalization ability for moderate problem sizes and indicate how freezing the Ansatz can dramatically reduce time-to-solution without degrading solution quality. The paper also discusses scalability limitations, the impact of ``warm-start'' initialization of parameters, and prospects for extension to more complex problems, such as Vehicle Routing and Job-Shop Scheduling.
This paper introduces a novel edge-based encoding technique for solving the Traveling Salesman Problem (TSP) on a quantum computer, reducing the required number of qubits. For implementation in real quantum devices, we applied the subspace reduction encoding to further reduce the dimension of the TSP solution space. We attack the TSP for 4-, 5-, and 6-city instances in both simulators and real quantum computers across different encoding frameworks. Optimal solutions of the 4-city TSP instance are obtained on state-of-the art IQM quantum computer. Our study presents a comparative analysis between edge-based encoding scheme and the node-based encoding methodology in the literature. Our findings indicate that the proposed encoding scheme outperforms conventional methods in terms of statistical measures, quantum resource utilization, and computational efficiency when applied to smaller TSP instances.
The Travelling Salesman Problem (TSP) is a well-known NP-hard combinatorial optimisation problem with important industrial applications. Although extensively studied, quantum solutions for instances beyond a dozen locations remain uncommon. We present a hybrid penalty-free Variational Quantum Algorithm (VQA) that significantly reduces both the qubit count and the runtime through a compact penalty-free formulation scaling as O(nlog2n)\mathcal{O}(nlog_2 n)O(nlog2​n) qubits compared to conventional formulations with O(n2)\mathcal{O}(n^2)O(n2) scaling. The VQA achieves solution qualities exceeding 90\% on Rigetti quantum hardware for instances of up to 17 locations. To the best of our knowledge, this represents the largest TSP instance solved on gate-based hardware. We systematically evaluate multiple encoding and optimisation strategies, and show that warm-start initialisation improves solution quality, while runtime is reduced by nearly two orders of magnitude through the use of gradient-free optimisation methods and cost-function caching. We additionally introduce a novel VQA inspired classical machine learning model (VQAi-CML) and benchmark the VQA against VQAi-CML, a greedy heuristic, a Monte Carlo baseline, and a strong classical optimisation baseline. Across the networks evaluated, the VQA consistently achieves superior solution quality relative to VQAi-CML, greedy heuristic, and Monte Carlo baselines, but not the classical solver
In this paper we present a variational algorithm for the Traveling Salesman Problem (TSP) thatcombines (i) a compact encoding of permutations, which reduces the qubit requirement too, (ii) anoptimize-freeze-reuse strategy: where the circuit topology (“Ansatz”) is first optimized on a train-ing instance by Simulated Annealing (SA), then “frozen” and re-used on novel instances, limitedto a rapid re-optimization of only the circuit parameters. This pipeline eliminates costly structuralresearch in testing, making the procedure immediately implementable on NISQ hardware.On a set of 40 randomly generated symmetric instances that span 4 − 7 cities, the resultingAnsatz achieves an average optimal trip sampling probability of 100% for 4 city cases, 90% for 5city cases and 80% for 6 city cases. With 7 cities the success rate drops markedly to an average of∼ 20%, revealing the onset of scalability limitations of the proposed method.The results show robust generalization ability for moderate problem sizes and indicate howfreezing the Ansatz can dramatically reduce time-to-solution without degrading solution quality.The paper also discusses scalability limitations, the impact of “warm-start” initialization of pa-rameters, and prospects for extension to more complex problems, such as Vehicle Routing andJob-Shop Scheduling.
Abstract The travelling salesman problem (TSP) is a popular NP-hard combinatorial optimization problem that requires finding the optimal way for a salesman to travel through different cities once and return to the initial city. The existing methods of solving TSPs on quantum systems are either gate-based or binary variable-based encoding. Both approaches are resource-expensive in terms of the number of qubits, while performing worse compared to existing classical algorithms, even for small-sized problems. A novel encoding scheme is needed to map the TSP problem onto a quantum system, which is addressed in this work. We introduce a distinct geometric approach to encode the TSP on a single qubit and present a quantum-inspired algorithm to solve the problem by invoking the principle of quantum superposition . The cities are represented as quantum states on the Bloch sphere, while the preparation of superposition states allows us to traverse multiple paths at once. The underlying framework of our algorithm is a quantum-inspired version of the classical Brachistochrone approach. Optimal control methods are employed to create a selective superposition of the quantum states to find the shortest route of a given TSP. The numerical simulations solve a sample of four to nine cities for which exact solutions are obtained. The algorithm can be implemented on any quantum platform capable of efficiently rotating a qubit and allowing state tomography measurements. For the TSP problem sizes considered in this work, our algorithm is more resource-efficient and accurate than existing quantum algorithms, with the potential for scalability.
The Traveling Salesman Problem (TSP) is a widely studied NP-complete optimization challenge with significant theoretical and practical implications. This study proposes a hybrid quantum-classical framework using a Quantum-Inspired Evolutionary Algorithm (QEA) with Sort Gray Binary Encoding to solve the TSP. The proposed method guarantees the generation of valid TSP tours by eliminating invalid solutions. It employs quantum superposition with intrinsic randomness to enhance computational efficiency and scalability. The framework was implemented on cloud-based NISQ platforms, including IBM Quantum and AWS Braket, demonstrating its practicality and effectiveness. Experimental evaluations revealed that the proposed framework successfully solved TSP instances with up to 15 cities, achieving superior performance compared to classical methods and showcasing its ability to scale under NISQ constraints. The results also highlight the potential of hybrid quantum-classical approaches to overcome hardware limitations in current quantum systems. This study establishes a robust hybrid methodology for solving combinatorial optimization problems. It also sets a benchmark for leveraging the capabilities of NISQ-era quantum devices in real-world applications, thereby providing a foundation for future research in hybrid quantum-classical optimization techniques.