Paper 2024/626

Exponential Quantum Speedup for the Traveling Salesman Problem

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 $NP$-hard with a brute-force complexity of $O(N^N)$ or $O(N^{2N})$ 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 $O(N^3\log(N)\kappa/\epsilon + 1/\epsilon^3)$, where the errors $\epsilon$ are $O(1/{\rm poly}(N))$, and $\kappa$ is the not-too-large condition number of the matrix encoding all Hamiltonian cycles.

Note: Added a sentence in the last section on why kappa cannot be exponentially large. Some more typos fixed too.