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

In this article, we present a 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 probability $100%$. In contrast, existing quantum search algorithms generally have 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 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.