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.
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.
The Traveling Salesman Problem (TSP) serves as a standard benchmark for assessing potential quantum advantage in combinatorial optimization. This paper provides a critical comparative analysis of quantum annealing, gate-based approaches based on quantum phase estimation, and hybrid quantum–classical dynamic programming methods within a unified complexity and implementation framework. While theoretical results indicate reductions of the classical O ( 2 n ) scaling to approximately O ( 1.728 n ) for general instances and to O ( 1.110 n ) for structured cases, these improvements remain largely asymptotic. In practice, all current quantum approaches are strongly constrained by hardware limitations. Quantum annealing suffers from embedding overhead and noise, hybrid methods rely extensively on classical decomposition without guaranteed optimality, and gate-based implementations are limited by circuit depth and error accumulation. As a result, no clear quantum advantage over classical algorithms has been demonstrated at practically relevant problem scales. By systematically separating intrinsic algorithmic scaling from hardware-induced resource constraints, this work clarifies the origin of the gap between theoretical quantum speedups and experimental performance, providing a critical assessment of the current status and realistic prospects of quantum TSP methods.
The Traveling Salesman Problem is a classical NP-hard problem with significant implications in logistics, circuit design, and operations research. This paper presents a comparative study of four approaches to solving the Traveling Salesman Problem: brute-force enumeration, a 2-approximation algorithm using minimum spanning trees, simulated annealing, and the Quantum Approximate Optimization Algorithm. We implement each technique and evaluate them on graphs of varying sizes to analyze performance, solution quality, and scalability. In doing so, we have also developed an open-source framework that allows researchers and practitioners to explore, test and extend these methods.
The famous Travelling Salesman Problem (TSP) is an important category of optimization problems that is mostly encountered in various areas of science and engineering. Studying optimization problems motivates to develop advanced techniques more suited to contemporary practical problems. Among those, especially the NP hard problems provide an apt platform to demonstrate supremacy of quantum over classical technologies in terms of resources and time. TSP is one such NP hard problem in combinatorial optimization which takes exponential time order for solving by brute force method. Here we propose a quantum algorithm to solve the travelling salesman problem using phase estimation technique. We approach the problem by encoding the given distances between the cities as phases. We construct unitary operators whose eigenvectors are the computational basis states and eigenvalues are various combinations of these phases. Then we apply phase estimation algorithm to certain eigenstates which give us all the total distances possible for all the routes. After obtaining the distances we can search through this information using the quantum search algorithm for finding the minimum to find the least possible distance as well the route taken. This provides us a quadratic speedup over the classical brute force method for a large number of cities. In this paper, we illustrate an example of the travelling salesman problem by taking four cities and present the results by simulating the codes in the IBM's quantum simulator.
Abstract The traveling salesman problem is the problem of finding out the shortest route in a network of cities, that a salesman needs to travel to cover all the cities, without visiting the same city more than once. This problem is known to be N P -hard with a brute-force complexity of O ( N N ) or O ( N 2 N ) for N number of cities. This problem is equivalent to finding out the shortest Hamiltonian cycle in a given graph, if at least one Hamiltonian cycle exists in it. Quantum algorithms for this problem typically provide with a quadratic speedup only, using Grover’s search, thereby having a complexity of O ( N N /2 ) or O ( N N ). We present a bounded-error quantum polynomial-time (BQP) algorithm for solving the problem, providing with an exponential speedup. The overall complexity of our algorithm is O ( N 3 log( N ) κ / ϵ + 1/ ϵ 3 ), where the errors ϵ are O (1/poly( N )), and κ is the not-too-large condition number of the matrix encoding all Hamiltonian cycles.
The Travelling Salesman Problem (TSP) is a classic combinatorial optimisation problem which is NP-hard in optimisation form and NP-complete as a decision problem. The standard question asks for the shortest tour that visits every city exactly once and returns to the origin. This paper studies a threshold-fraction variant, which we call the robust TSP, formulated as a promise problem: given a distance threshold K and a promised fraction p, decide whether at least a fraction p of all tours have total length at most K, or whether no such tour exists. We show that Grover's quantum search algorithm decides this problem using O(1/p) oracle queries, independent of the number of cities N, with each query implemented by a quantum circuit of polynomial size in N. For any fixed robustness level p, the total runtime is therefore polynomial in N. Measured in oracle queries, the quantum algorithm offers a quadratic improvement in the dependence on 1/p over classical random sampling, which needs Θ(1/p) samples; because each quantum query is costly, however, it is faster in total running time only when p is very small (roughly below N−4), where the number of queries is no longer constant. We describe the problem formulation, detail the oracle construction (edge-length loading, sequential quantum addition, and threshold comparison), and analyse the query complexity to achieve the required result.
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.