Walking on Vertices and Edges by Continuous-Time Quantum Walk

TL;DR

This paper introduces a novel continuous-time quantum walk model allowing transitions between vertices and edges, and demonstrates its effectiveness in spatial search algorithms on bipartite graphs with optimal running time.

Contribution

It defines a new quantum walk framework that includes vertex-edge transitions and applies it to improve spatial search algorithms on bipartite graphs.

Findings

Optimal search time of O(√N_e) for vertices or edges

Success probability approaches 1 for large graphs

Extension of quantum walk models to include edge transitions

Abstract

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.