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

Since Grover devised a quantum algorithm for unstructured search, generalization of the algorithm to structured data sets represented by graphs has been an important research topic. The introduction of absorbing marked vertices provided a breakthrough for this problem, and recently it was proved that a quantum walk search algorithm replacing the marked vertices by partially absorbing vertices can find a marked vertex in any reversible Markov chain with any number of marked vertices. However, in contrast, the proof based on the quantum fast-forwarding technique gives little intuition about the underlying mechanism, while the spectral analysis of Grover’s algorithm leads to understanding of the searching mechanism as a rotation in a two-dimensional space. For a spectral approach to the quantum search on Markov chains, we consider as a nontrivial example the complete bipartite graph consisting of two sets X1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1$$\end{document} and X2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_2$$\end{document} and the marked vertices being only in X2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_2$$\end{document}. By analytically determining the spectral information of the quantum walk, we demonstrate that the quantum algorithm shows quadratic speed-up compared to the corresponding classical search method. And we find that the quantum search is described in terms of a two-state model for that case.