Hybrid quantum-classical algorithms are the subject of intensive research in quantum computing and specifically in the domains of Machine Learning and Approximate Optimization. Simple distance-based learning classifiers such as kNN or k-means classifiers have already been the subject of many proposals, but most of their quantum implementation only consider a very small number of classes/clusters and only use metrics suitable for normalized unit length quantum data. Very few approaches combine the need for Euclidean distance of unnormalized data with an efficient quantum computation implementation. Based on two kNN and k-means proposals from the literature, this paper proposes their adaptation to the design of a Quantum Self-Organizing Map in order to solve the NP-hard Traveling Salesman Optimization Problem (TSP) specifically. Hence, we evaluate the hybrid quantum-classical approach to Euclidean TSP instances of sizes from 52 to 127 cities and present the properties, advantages, and limitations of the quantum implementation in solving the TSP optimization problem.
Hybrid quantum-classical algorithms are the subject of intensive research in quantum computing and specifically in the domains of Machine Learning and Approximate Optimization. Simple distance-based learning classifiers such as kNN or k-means classifiers have already been the subject of many proposals, but most of their quantum implementation only consider a very small number of classes/clusters and only use metrics suitable for normalized unit length quantum data. Very few approaches combine the need for Euclidean distance of unnormalized data with an efficient quantum computation implementation. Based on two kNN and k-means proposals from the literature, this paper proposes their adaptation to the design of a Quantum Self-Organizing Map in order to solve the NP-hard Traveling Salesman Optimization Problem (TSP) specifically. Hence, we evaluate the hybrid quantum-classical approach to Euclidean TSP instances of sizes from 52 to 127 cities and present the properties, advantages, and limitations of the quantum implementation in solving the TSP optimization problem.
The Traveling Salesperson Problem (TSP) is a fundamental NP-hard optimisation challenge with widespread applications in logistics, operations research, and network design. While classical algorithms effectively solve small to medium-sized instances, they struggle with scalability due to exponential complexity. In this work, we present a hybrid quantum-classical approach that leverages IBM's Qiskit Runtime to integrate quantum optimisation techniques with classical machine learning methods, specifically K-Means clustering and Random Forest classifiers. These machine learning components aid in problem decomposition and noise mitigation, improving the quality of quantum solutions. Experimental results for TSP instances ranging from 4 to 8 cities reveal that the quantum-only approach produces solutions up to 21.7% worse than the classical baseline, while the hybrid method reduces this cost increase to 11.3% for 8 cities. This demonstrates that hybrid approaches improve solution quality compared to purely quantum methods but remain suboptimal compared to classical solvers. Despite current hardware limitations, these results highlight the potential of quantum-enhanced methods for combinatorial optimisation, paving the way for future advancements in scalable quantum computing frameworks.
In this paper, we design, analyze, and evaluate a hybrid quantum algorithm for the metric traveling salesman problem (TSP). TSP is a well-studied NP-complete problem that many algorithmic techniques have been developed for, on both classic computers and quantum computers. The existing literature of algorithms for TSP are neither adaptive to input data nor suitable for processing medium-size data on the modern classic and quantum machines. In this work, we leverage the classic computers’ power (large memory) and the quantum computers’ power (quantum parallelism), based on the input data, to fasten the hybrid algorithm’s overall running time. Our algorithmic ideas include trimming the input data efficiently using a classic algorithm, finding an optimal solution for the post-processed data using a quantum-only algorithm, and constructing an optimal solution for the untrimmed data input efficiently using a classic algorithm. We conduct experiments to compare our hybrid algorithm against the state-of-the-art classic and quantum algorithms on real data sets. The experimental results show that our solution truly outperforms the others and thus confirm our theoretical analysis. This work provides insightful quantitative tools for people and compilers to choose appropriate quantum or classical or hybrid algorithms, especially in the NISQ (noisy intermediate-scale quantum) era, for NP-complete problems such as TSP.
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 Traveling Salesman Problem (TSP), a well-known NP-hard optimization problem, has been widely studied for decades due to its broad practical applicability. In recent years, deep learning methods have shown promise in solving TSP; however, existing approaches often struggle to extract informative edge features between cities, which limits solution quality. To address this, we propose a Quantum-Inspired Deep Learning (QIDL) model that enhances node representations through quantum-inspired embedding, integrates Euclidean distance into the Transformer to strengthen spatial reasoning, and employs a quantum-inspired pointer network to replace greedy decoding with global optimization search. Experimental results demonstrate that QIDL outperforms traditional pointer networks, achieving faster convergence, improved stability, and a 1.76% reduction in tour length, with strong performance on medium-scale TSP instances. These results highlight the potential of QIDL to open new avenues for quantum-inspired learning in combinatorial optimization.
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.
This paper implements a new way of solving a problem called the traveling salesman problem (TSP) using quantum genetic algorithm (QGA). We compared how well this new approach works to the traditional method known as a classical genetic algorithm (CGA). The TSP is a well-established challenge in combinatorial optimization where the objective is to find the most efficient path to visit a series of cities, minimizing the total distance, and returning to the starting point. We chose the TSP to test the performance of both algorithms because of its computational complexity and importance in practical applications. We choose the dataset from the international standard library TSPLIB for our experiments. By designing and implementing both algorithms and conducting experiments on various sizes and types of TSP instances, we provide an in-depth analysis of the accuracy of the optimal solution, the number of iterations, the execution time, and the stability of the algorithms for both. The empirical findings indicate that the CGA outperforms the QGA in terms of finding superior solutions more quickly in most of the test instances, especially when the problem size is large. This suggests that although the principle of quantum computing provides a new way to solve complex combinatorial optimisation problems, the implementation of quantum phenomena and the setting of parameters such as the optimal angle for a quantum revolving gate is challenging and need further optimisation to achieve the desired results. Additionally, it is important to note that the QGA has not been tested on real quantum hardware, so its true performance remains unverified. These limitations provide rich opportunities for further research in the future.