Impacts of Multiple Solutions on the Lackadaisical Quantum Walk Search Algorithm

Abstract

The lackadaisical quantum walk is a graph search algorithm for 2D grids whose vertices have a self-loop of weight l. Since the technique depends considerably on this l, research efforts have been estimating the optimal value for different scenarios, including 2D grids with multiple solutions. However, specifically two previous works have used different stopping conditions for the simulations. Here, firstly, we show that these stopping conditions are not interchangeable. After doing such a pending investigation to define the stopping condition properly, we analyze the impacts of multiple solutions on the final results achieved by the technique, which is the main contribution of this work. In doing so, we demonstrate that the success probability is inversely proportional to the density of vertices marked as solutions and directly proportional to the relative distance between solutions. These relations presented here are guaranteed only for high values of the input parameters because, from different points of view, we show that a disturbed transition range exists in the small values.