Quantum-walk-based search algorithms with multiple marked vertices

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, with 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.

Automated summary

This research addresses the problem of calculating the time complexity of finding a marked vertex using quantum-walk-based search algorithms on arbitrary graphs, extending previous analytical methods which were limited to bipartite graphs. Quantitative examples based on two-dimensional lattices and hypercubes demonstrate the details of the method.