Scaling in the dynamics of directed lackadaisical quantum walks

Abstract

We study transport properties of directed lackadaisical quantum walks. In a lackadaisical walk, in addition to the possibility of moving out of a node, the walker can remain on the same node with some probability. This is achieved by introducing self-loops, parameterized by self-loop strength l, attached to the nodes such that large l implies a higher likelihood for the walker to be trapped at the node. By analytically calculating the mean walker position, we demonstrate the existence of two distinct scaling regimes with l for walks on a line and on a binary tree. Furthermore, we show that by tuning the initial state, the dynamics of the quantum walker can be manipulated.