Deterministic spatial search using alternating quantum walks

Abstract: This paper examines the performance of spatial search where the Grover diffusion operator is replaced by continuous-time quantum walks on a class of interdependent networks. We prove that for a set of optimal quantum walk times and marked vertex phase shifts, a deterministic algorithm for structured spatial search is established that finds the marked vertex with 100% probability. This improves on the original spatial search algorithm on the same class of graphs, which we show can only amplify to 50% probability. Our method uses $\left\lceil\frac{\pi}{2\sqrt{2}}\sqrt{N}\right\rceil$ marked vertex phase shifts for an $N$-vertex graph, making it comparable with Grover's algorithm for unstructured search. It is expected that this new framework can be readily extended to deterministic spatial search on other families of graph structures.