A quantum speedup algorithm for TSP based on quantum dynamic programming with very few qubits

TL;DR

This paper presents a quantum algorithm that efficiently prepares a superposition of all TSP solutions using minimal qubits, enabling exponential speedup in quantum search for solving TSP.

Contribution

It introduces a novel quantum dynamic programming method to generate uniform superpositions of Hamiltonian cycles with very few qubits, achieving exponential acceleration.

Findings

Achieves polynomial gate complexity for superposition preparation.

Realizes the theoretical minimum query complexity for quantum TSP search.

Provides a practically implementable quantum algorithm for TSP.

Abstract

The Traveling Salesman Problem (TSP) is a classical NP-hard problem that plays a crucial role in combinatorial optimization. In this paper, we are interested in the quantum search framework for the TSP because it has robust theoretical guarantees. However, we need to first search for all Hamiltonian cycles from a very large solution space, which greatly weakens the advantage of quantum search algorithms. To address this issue, one can first prepare a superposition state of all feasible solutions, and then amplify the amplitude of the optimal solution from it. We propose a quantum algorithm to generate the uniform superposition state of all N-length Hamiltonian cycles as an initial state within polynomial gate complexity based on pure quantum dynamic programming with very few ancillary qubits, which achieves exponential acceleration compared to the previous initial state preparation algorithm. As a result, we realized the theoretical minimum query complexity of quantum search algorithms for a general TSP. Compared to some algorithms that theoretically have lower query complexities but lack practical implementation solutions, our algorithm has feasible circuit implementation. Our work provides a meaningful research case on how to fully utilize the structures of specific problems to unleash the acceleration capability of the quantum search algorithms.