Lackadaisical quantum walks on 2D grids with multiple marked vertices

Abstract

Lackadaisical quantum walk (LQW) is a quantum analog of a classical lazy walk, where each vertex has a self-loop of weight l. For a regular 2D grid LQW can find a single marked vertex with O(1) probability in steps using l = d/N, where d is the degree of the vertices of the grid []. For multiple marked vertices, however, l = d/N is not optimal as the success probability decreases with the increase of the number of marked vertices []. In this paper, we numerically study search by LQW for different types of 2D grids—triangular, rectangular and honeycomb—with multiple marked vertices. We show that in all cases the weight l = m ⋅ d/N, where m is the number of marked vertices, still leads to O(1) success probability.