Spatial search algorithms on graphs with multiple targets using discrete-time quantum walk

Abstract

Search algorithms based on discrete-time quantum walk (QW) can be considered as alterations of the standard QW: Use a different coin operator that distinguishes target and nontarget vertices, or, mark the target vertices first followed by the standard QW. Two most frequently used marking coins are [Formula: see text] and [Formula: see text] the negative identity operator and the negative Grover diffusion operator. We show that search algorithms corresponding to these four combinations can be reduced to two, denoted as [Formula: see text] and [Formula: see text], and they are equivalent when searching for nonadjacent multiple targets. For adjacent target vertices, numerical simulations show that the performance of the algorithm [Formula: see text] highly depends on the density of the underlying graph, and it outperforms [Formula: see text] when the density is large enough. At last, a generalized stationary state of both search algorithms on the graphs with even-numbered degree is provided.