Lackadaisical quantum walks on 2D grids with multiple marked vertices

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 √N × √N 2D grid LQW can find a single marked vertex with O(1) probability in O(√N log N) steps using l = d/N, where d is the degree of the vertices of the grid [11]. 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 [12]. 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.