Unifying quantum spatial search, state transfer and uniform sampling on graphs: simple and exact

TL;DR

This paper introduces a unified quantum walk framework that achieves exact uniform sampling, perfect state transfer, and deterministic spatial search on various graphs, improving previous results and offering new capabilities.

Contribution

The paper presents a simple, exact quantum walk-based framework unifying multiple quantum algorithms on graphs, with broad applicability and improved performance.

Findings

Achieves exact uniform sampling over all vertices.

Enables perfect state transfer between any two vertices.

Provides deterministic spatial search on vertex-transitive graphs.

Abstract

This article presents a novel and succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer and uniform sampling on a large class of graphs. Using the framework, we can achieve exact uniform sampling over all vertices and perfect state transfer between any two vertices, provided that eigenvalues of Laplacian matrix of the graph are all integers. Furthermore, if the graph is vertex-transitive as well, then we can achieve deterministic quantum spatial search that finds a marked vertex with certainty. In contrast, existing quantum search algorithms generally has a certain probability of failure. Even if the graph is not vertex-transitive, such as the complete bipartite graph, we can still adjust the algorithmic framework to obtain deterministic spatial search, which thus shows the flexibility of it. Besides unifying and improving plenty of previous results, our work provides new results on more graphs. The approach is easy to use since it has a succinct formalism that depends only on the depth of the Laplacian eigenvalue set of the graph, and may shed light on the solution of more problems related to graphs.