An Efficient Quantum Algorithm for the Traveling Salesman Problem
Year: 2025
Type: article
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.
Authors: Anant Sharma, Nupur Deshpande, Sanchita Ghosh, Sreetama Das, Shibdas Roy
Institutions: Université de Bourgogne, TCG Crest, Indian Institute of Science Education and Research, Tirupati, Delft University of Technology, European Laboratory for Non-Linear Spectroscopy, Institute for Cross-Disciplinary Physics and Complex Systems, National Institute of Optics, University of Florence, Universitat de les Illes Balears, Academy of Scientific and Innovative Research
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4411665597