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 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 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.
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.
Quantum-Inspired Evolutionary Algorithms (QIEAs) represent a novel fusion of quantum computing principles and classical evolutionary strategies, offering a powerful approach to solving complex optimization problems. This chapter provides a comprehensive overview, detailing foundational concepts, implementation techniques, and potential applications of QIEAs. By leveraging quantum phenomena such as superposition and entanglement, QIEAs enhance the exploitation and exploration capabilities of traditional evolutionary algorithms (EAs), thus enabling an efficient search of vast solution spaces. The study examines key components of QIEA, including quantum encoding, rotation gates, and entanglement operations, and compares its performance with classical EAs. By analyzing real-life examples of combinatorial optimization problems (COPs) such as the minimum spanning tree (MST), traveling salesman problem (TSP), and graph coloring, the chapter demonstrates QIEA’s superior efficiency and solution quality. Despite implementation challenges, QIEA shows promise in various domains, including machine learning, finance, healthcare, telecommunications, and e-commerce. Future research directions are outlined, emphasizing the need for simplified quantum-inspired operators, hybrid quantum-classical systems, and interdisciplinary collaboration. This chapter highlights the transformative potential of QIEA, setting the stage for further advancements and applications in optimization and computational efficiency.
We present novel path-slicing strategies integrated with quantum local search to optimize solutions for the Traveling Salesman Problem (TSP), addressing the limitations of current Noisy Intermediate-Scale Quantum (NISQ) technologies. Our hybrid quantum-classical approach leverages classical path initialization and quantum optimization to effectively manage the computational challenges posed by the TSP. We explore various path slicing methods, including k-means and anti-k-means clustering, to divide the TSP into manageable subproblems. These are then solved using quantum or classical solvers. Our analysis, performed on multiple TSP instances from the TSPlib, demonstrates the ability of our strategies to achieve near-optimal solutions efficiently, highlighting significant improvements in solving efficiency and resource utilization. This approach paves the way for future applications in larger combinatorial optimization scenarios, advancing the field of quantum optimization.
A hybrid quantum-classical framework is exploited to solve the Traveling Salesman Problem (TSP) based on a Quantum Approximate Optimization Algorithm (QAOA). To overcome the computational limitation of the Grover’s Search Algorithm (GSA) stemming from path encoding and arithmetic subroutines, a clustering-based strategy is employed to decompose a 16-city TSP into four 4-city subproblems, each of which can be solved with a 16-qubit budget. Intra-cluster solutions are optimized with QAOA, and are classically stitched to construct the global tour. Our results clearly validate the effectiveness of the hierarchical design of the algorithm, highlighting its feasibility for mid-scale quantum optimizations.
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.