Walking on vertices and edges by continuous-time quantum walk

The article delves into the intricacies of continuous-time quantum walks, which are the quantum counterparts of random walks. It explores the dynamics of these walks, governed by the laws of quantum mechanics with an additional locality constraint. The spatial structure of quantum walks can vary, including vertices, edges, and even more complex structures like triangles and embeddings. The focus is on the continuous-time quantum walk model, which enables the walker to hop between vertices and edges, a task previously found challenging with discrete-time quantum walks. The spatial search algorithm by continuous-time quantum walks is highlighted, with experimental implementations and theoretical analyses. The total graph of a graph G, which associates vertices with both vertices and edges, plays a crucial role in defining this model. The article analyzes the computational complexity of search algorithms on graphs, particularly the complete bipartite graph, demonstrating that the optimal running time to find a marked vertex or edge is achievable with a high success probability. The method to determine the computational complexity is rigorously outlined, along with numerical checks that validate the hypotheses used. The work concludes with potential extensions of the total quantum walk model, inviting further exploration in future research.