Lackadaisical quantum walks on 2D grids with multiple marked vertices

TL;DR

This paper investigates lackadaisical quantum walks on various 2D grid types with multiple marked vertices, demonstrating that an adjusted self-loop weight maintains a constant success probability across different grid structures.

Contribution

It extends the understanding of lackadaisical quantum walks to multiple marked vertices on various 2D grids, proposing a new weight scaling for improved search success.

Findings

Success probability remains constant with weight l = m·d/N across grid types.

LQW effectively finds multiple marked vertices with adjusted weight.

Different 2D grid structures do not affect the success probability when using the proposed weight.

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 $\sqrt{N}\times\sqrt{N}$ 2D grid LQW can find a single marked vertex with $O(1)$ probability in $O(\sqrt{N\log N})$ 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 $l = m\cdot d/N$, where $m$ is the number of marked vertices, still leads to $O(1)$ success probability.