Quantum speedup of the Travelling Salesman Problem for bounded-degree graphs

TL;DR

This paper demonstrates a quadratic quantum speedup for solving the Traveling Salesman Problem on bounded-degree graphs, specifically for degrees up to 3 and 4, by applying quantum backtracking techniques.

Contribution

It introduces quantum algorithms that significantly accelerate classical solutions for bounded-degree TSP instances, extending quantum speedup to degrees up to 4.

Findings

Quadratic quantum speedup for degree-3 graphs

Speedup for degree-4 graphs via reductions

Extension of quantum techniques to higher degrees

Abstract

The Travelling Salesman Problem is one of the most famous problems in graph theory. However, little is currently known about the extent to which quantum computers could speed up algorithms for the problem. In this paper, we prove a quadratic quantum speedup when the degree of each vertex is at most 3 by applying a quantum backtracking algorithm to a classical algorithm by Xiao and Nagamochi. We then use similar techniques to accelerate a classical algorithm for when the degree of each vertex is at most 4, before speeding up higher-degree graphs via reductions to these instances.