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arXiv:2512.17291

An edge-based and subspace reduction encoding scheme to solve the traveling salesman problem in quantum computers

Anandu Kalleri Madhu, Chi-Kwong Li, Jami Rönkkö, Mikio Nakahara, Ray-Kuang Lee

quant-phSubmitted Dec 19, 2025• Announced Dec 22, 2025• 0 citations
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Abstract

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.

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14 pages, 6 figures, 3 tables

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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 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.
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.
In this paper, we consider a method for solving difficult combinatorial optimization problems on real quantum computers. We focus on the traveling salesman problem as a representative problem for a group of problems where the solution is represented by a permutation. Typically, existing algorithmic solutions use binary matrices to store this permutation – the QUBO (quadratic unconstrained binary optimization) model. We propose a new way of encoding permutations on quantum computers, using a significantly smaller number of qubits than binary matrix encodings. Our method allows for significant performance improvements for any problem whose input or solution is a permutation. We demonstrate an example implementation of the traveling salesman problem on the IQM quantum computers: IQM Spark 5-qubit ‘ODRA-5’ computer and IQM Radiance ‘Garnet’ 20-qubit computer.
The Traveling Salesman Problem (TSP) is a classical NP-hard problem in combinatorial optimization, where determining the shortest route among a set of cities becomes computationally prohibitive as the problem size increases. This work explores quantum computing as an alternative approach to address this complexity. Unlike existing methods that primarily rely on quantum annealing, we propose a quantum iterative framework integrating Quantum Phase Estimation (QPE) and Grover's search algorithm. Route costs are encoded as quantum phases, enabling QPE to efficiently evaluate them, while Amplitude Amplification, implemented via the Grover-Long algorithm, iteratively refines the solution space toward the optimal route. A proof-of-concept case study on a small-scale TSP instance demonstrates the feasibility of this approach and its potential for scaling to larger optimization problems. Furthermore, under an expectation-based analysis, the algorithm exhibits an expected computational complexity of O(m2log⁡2(m)log⁡2(1/ε)ε)O(\frac{m^2\log_2(m)\log_2(1/ε)}{\sqrtε})O(ε​m2log2​(m)log2​(1/ε)​) which depends on the error tolerance parameter εεε. This estimation omits the initialization term, which we expect future refinements to render subdominant to Phase Estimation.
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.
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.