The sojourn times of one dimensional discrete-time quantum walks

Abstract

In the existing literature, a sojourn time of a discrete-time quantum walk is not a random variable. To solve this problem, we redefine the sojourn time of a quantum walk where its coin evolution operator can be general. We first discuss a class of quantum walks governed by flip operators. We cumulatively calculate how much time a walker spends in the set of non-negative integers up to a fixed evolution time. Whether a walker makes a left or right evolution, we add up the staying times as long as it stays within the target set. We define a sojourn time as the total amount of the staying times. Compared with existing definitions, we show that this definition can satisfy the probability normalization. From this, we define a random variable about the sojourn time and discuss its probability distribution. We build a mathematical model to characterize a sojourn time that is embedded into a quantum walk. These results are also valid for a class of quantum walks governed by general coin operators. We also give a method for calculating the sojourn time and analyze the shape features of its probability distribution.