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.
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.
This paper presents a new algorithm for solving the travelling salesman problem (TSP). The TSP is one of the most known combinatorial optimisation problems. It is about finding the shortest Hamiltonian cycle relating N cities. The algorithm is inspired from both genetic algorithms and quantum computing fields. It extends the standard genetic algorithms by combining them to some concepts and principles provided from quantum computing field such as quantum bit, states superposition and interference. The obtained results from the application of the proposed algorithm on some instances of TSP are significantly better than those provided by standard genetic algorithms.
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.
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.
Optimization algorithms for combinatorial problems play a critical role in a wide range of applications, from logistics and scheduling to cryptography and artificial intelligence. Traditional classical methods, such as simulated annealing, genetic algorithms, and branch-and-bound techniques, have been widely employed to solve these problems, but they often face limitations in terms of computational efficiency and scalability as problem complexity grows. In recent years, quantum computing has emerged as a promising alternative, with quantum annealing offering a novel approach to solving combinatorial optimization problems. This study provides a comparative analysis of quantum annealing and classical optimization methods, highlighting their respective strengths and limitations. We explore how quantum annealing, through leveraging quantum tunnelling and superposition, has the potential to outperform classical algorithms in specific problem domains and Noisy Mean Field Annealing (NMFA) is used to comparisons of all algorithms. Benchmarking various combinatorial problems, such as the Traveling Salesman Problem (TSP), Max-Cut, Knapsack, and Quadratic Assignment Problem (QAP), this paper discusses performance metrics, computational time, and scalability. The findings suggest that while classical methods remain robust and practical for many real-world problems, quantum annealing holds significant promise, especially as quantum hardware continues to mature. However, there remain challenges in terms of noise, coherence, and problem mapping, which currently limit the full realization of quantum annealing's potential. This study offers insights into the future directions of optimization techniques and the evolving role of quantum computing in solving complex combinatorial problems.
Quantum genetic algorithm(QGA) was proved to be better than the conventional genetic algorithms on numerical and combinational optimization problems,but it is usually used to solve the knapsack problems of the combinational optimization.It is also possible to use its strong ability of exploitation and exploration to other difficult problems,for example,the TSP,a class of NP-hard combinatorial optimization problems.First,a new encoding scheme with pairs of amplitudes is designed.A quantum individual is corresponding to a vector,and the vector is corresponding to the unique valid tour,and vice versa.The advantages of the encoding scheme are that it always generates feasible solution,uses less population size and storage,and can effectively enhance the diversity of the population.Second,the quantum crossover can maintain the relatively good gene blocks.Third,the two-phase local search for edge evolving is integrated into QGA to accelerate its convergent speed.The first phase is used to optimize the sparsely located cities,and the second phase is used to optimize densely located cities.Based on these,a novel and efficient QGA for TSP is proposed,and its convergence to global optimal solution with probability one is proved.The numerical experiments show that the proposed algorithm can find the global optimal solution with less computation and evolving time.
The Travelling Salesman Problem (TSP) asks for the shortest tour that visits each city once and returns to the start. It is a classic NP-hard problem and a benchmark for optimization, logistics, and network design. This paper surveys classical, quantum, and hybrid algorithms for the TSP using clear, step-by-step explanations. We summarize how each method works, what it guarantees, and how it scales with the number of cities. We also discuss practical limits of current quantum hardware and outline open problems and promising research directions. While quantum techniques can speed up certain subroutines or offer strong heuristics, no known method solves general TSP in polynomial time. Our goal is to offer a readable yet rigorous map of the landscape for students and practitioners.