Spectral approach to quantum searching on the interpolated Markov chains: the complete bipartite graph

Since Grover developed a quantum search algorithm for unstructured data, generalizing the algorithm to structured datasets, which can be represented as graphs, has been an important research topic. Recently, Krovi et al. introduced a quantum algorithm that can find a marked vertex on any ergodic and reversible graph by using absorbing marked vertices, replacing the marked vertices with partially absorbing vertices. The algorithm was extended to finding marked vertices in any graph with multiple marked vertices using the quantum fast-forwarding technique (QFF). However, the proof of this result based on QFF does not provide much intuition about the underlying mechanism. In this paper, to obtain the underlying mechanism of the quantum search with absorbing marked vertices, we consider, as a nontrivial example, the complete bipartite graph consisting of two sets \(X_1\) $X_1$ and \(X_2\) $X_2$ with the marked vertices being on both vertex sets. By analytically finding the spectral information of the quantum walk, we propose an approximate optimal search parameter \(s_o\) $s_o$ . We demonstrate that the quantum algorithm at the optimal search parameter \(s_o\) $s_o$ shows a quadratic speedup compared to the corresponding classical search method if the number of marked vertices is smaller than the total number of vertices. Additionally, we find that the quantum search is described in terms of a two-state model for that case.