Walking on Vertices and Edges by Continuous-Time Quantum Walk

The quantum walk dynamics obey the laws of quantum mechanics with an extra locality constraint, which demands that the evolution operator is local in the sense that the walker must visit the neighboring locations before endeavoring to distant places. Usually, the Hamiltonian is obtained from either the adjacency or the laplacian matrix of the graph and the walker hops from vertices to neighboring vertices. In this work, we define a version of the continuous-time quantum walk that allows the walker to hop from vertices to edges and vice versa. As an application, we analyze the spatial search algorithm on the complete bipartite graph by modifying the new version of the Hamiltonian with an extra term that depends on the location of the marked vertex or marked edge, similar to what is done in the standard continuous-time quantum walk model. We show that the optimal running time to find either a vertex or an edge is $O(\sqrt{N_e})$ with success probability $1+o(1)$, where $N_e$ is the number of edges of the complete bipartite graph.