  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.
