Lackadaisical quantum walks with multiple marked vertices

TL;DR

This paper investigates lackadaisical quantum walks with self loops on a 2D grid with multiple marked vertices, revealing limitations and proposing adjustments to improve search efficiency.

Contribution

It identifies exceptional configurations and optimizes self-loop weights for better quantum search performance with multiple marked vertices.

Findings

Existence of exceptional configurations with no quantum speed-up

Current self-loop weights are suboptimal for multiple marked vertices

Adjusted self-loop weights improve search efficiency

Abstract

The concept of lackadaisical quantum walk -- quantum walk with self loops -- was first introduced for discrete-time quantum walk on one-dimensional line. Later it was successfully applied to improve the running time of the spacial search on two-dimensional grid. In this paper we study search by lackadaisical quantum walk on the two-dimensional grid with multiple marked vertices. First, we show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configuration, i.e. placements of marked vertices for which the walk has no speed-up over the classical exhaustive search. Next, we demonstrate that the weight of the self-loop suggested in the previous papers is not optimal for multiple marked vertices. And, last, we show how to adjust the weight of the self-loop to overcome the aforementioned problem.