Quantum walk-based search algorithms with multiple marked vertices

TL;DR

This paper develops analytical methods to calculate the time complexity of quantum walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous bipartite graph limitations.

Contribution

It introduces a new analytical approach for quantum walks on general graphs, beyond bipartite cases, demonstrated through 2D lattices and hypercubes.

Findings

Analytical expressions for search time complexity on arbitrary graphs.

Extension of Szegedy's quantum walk methods to non-bipartite graphs.

Application to 2D lattices and hypercubes shows the method's versatility.

Abstract

The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.