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Quantum Information and Computation

An Efficient Quantum Algorithm for the Traveling Salesman Problem

Anant Sharma, Nupur Deshpande, Sanchita Purohit Ghosh, Sreetama Das, Shibdas Roy

Published Feb 22, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.2478/qic-2025-0004

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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 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.
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) 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.
We propose a quantum heuristic algorithm to solve the traveling salesman problem by generalizing the Grover search. Sufficient conditions are derived to greatly enhance the probability of finding the tours with the cheapest costs reaching almost to unity. These conditions are characterized by the statistical properties of tour costs and are shown to be automatically satisfied in the large-number limit of cities. In particular for a continuous distribution of the tours along the cost, we show that the quantum heuristic algorithm exhibits a quadratic speedup compared to its classical heuristic algorithm.
The traveling salesperson problem (TSP) is “if a traveling salesman wishes to visit exactly once each of a list ofmmmcities (where the cost of traveling from cityiiito cityjjjiscijc_{ij}cij​) and then return to the home city, what is the least costly route the traveling salesman can take [1]?” The TSP is an NP-Complete problem. This paper introduces a solution to the TSP using Grover's quantum algorithm. This paper also introduces a quantum algorithm to the minimum (and maximum) spanning tree problem. The full circuit implementation is presented using the open-source Qiskit environment [2]. This paper provides the number of qubits required, the quantum depth, and the quantum volume of the algorithm for the TSP. This paper also shows how to solve the minimum (and maximum) spanning tree using Grover's algorithm.