Faster search of clustered marked states with lackadaisical quantum walks

The nature of discrete-time quantum walks in the presence of multiple marked states can be found in the literature. An exceptional configuration of clustered marked states, which is a variant of multiple marked states, may be defined as a cluster of k marked states arranged in a k×k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{k} \times \sqrt{k}$$\end{document} array within a N×N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{N} \times \sqrt{N}$$\end{document} grid, where k=n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=n^{2}$$\end{document} and n an odd integer. In this article, we establish through numerical simulation that for lackadaisical quantum walks, which is the analogue of a three-state discrete-time quantum walks on a line, the success probability to find a vertex in the marked region of this exceptional configuration is nearly 1 with smaller run-time. We also show that the weights of the self-loop suggested for multiple marked states in the state-of-the-art works are not optimal for this exceptional configuration of clustered marked states. We propose a weight of the self-loop which gives the desired result for this configuration.