Quantum heuristic algorithm for traveling salesman problem

TL;DR

This paper introduces a quantum heuristic algorithm based on Grover search to efficiently solve the traveling salesman problem, achieving near-certain success probabilities and quadratic speedup for Gaussian-distributed tour costs.

Contribution

It generalizes Grover search to optimize TSP solutions and derives conditions for high success probability, demonstrating quadratic speedup for Gaussian cost distributions.

Findings

Achieves near-unity success probability for optimal tours

Demonstrates quadratic speedup over classical algorithms for Gaussian costs

Provides statistical characterization of tour costs for quantum search enhancement

Abstract

We propose a quantum heuristic algorithm to solve a traveling salesman problem by generalizing Grover search. Sufficient conditions are derived to greatly enhance the probability of finding the tours with extremal costs, reaching almost to unity and they are shown characterized by statistical properties of tour costs. In particular for a Gaussian distribution of the tours along the cost we show that the quantum algorithm exhibits the quadratic speedup of its classical counterpart, similarly to Grover search.