# One-dimensional lackadaisical quantum walks

Kun WANG, Nan Wu, Ping Xu, Fangmin Song

November 2017
research### Abstract

In this paper, we study the properties of lackadaisical quantum walks (LQWs) on a line. This model was first proposed in (Wong et al., J. Phys. A: Math. Theor. 48 435304, 2015) as a quantum analogue of lazy random walks where each vertex is attached to τ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate the wave function of the walker starting from the origin and obtain a long-time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary τ, and give an analytic solution for the variance of the walker’s probability distribution.

Publication

*Journal of Physics A: Mathematical and Theoretical*