Spatial Search for Two Marked Vertices on Hypercube by Continuous-Time Quantum Walk

Abstract

Search problem have a wide range of applications both in classical and quantum computers. In this work, the spatial search for a single marked vertex by continuous-time quantum walk (CTQW) is generalized to the search for multiple marked vertices. For many kinds of graphs with symmetrical structure, such as hypercube graph, the search for arbitrary single marked vertex is equivalent. However, this is not true for the search of multiple marked vertices and the search time is depend on the relative location of the marked vertices. We first give the spectrum and eigenspace of hypercube by using the theory of Cartesian product of graphs. Then, with the knowledge of spectrum and eigenspace, we analytical present the spatial search for all different configurations, namely all possible Hamming distance, of two marked vertices on hypercube. We find that although the different Hamming distance lead to unequal search time, this search can be done in \(\mathrm{O}\left( {\sqrt{N} } \right) \) time for all two uniform marked vertices.