Stopping time of a one-dimensional bounded quantum walk**Project supported by the National Natural Science Foundation of China (Grant Nos. 11222430, 11434011, and 11474049), the National Basic Research Program of China (Grant No. 2012CB922104), the Fundamental Research Funds for the Central Universities, China, and the Research Funds of Renmin University of China (Grant No. 16XNLQ03).

Abstract

The stopping time of a one-dimensional bounded classical random walk (RW) is defined as the number of steps taken by a random walker to arrive at a fixed boundary for the first time. A quantum walk (QW) is a non-trivial generalization of RW, and has attracted a great deal of interest from researchers working in quantum physics and quantum information. In this paper, we develop a method to calculate the stopping time for a one-dimensional QW. Using our method, we further compare the properties of stopping time for QW and RW. We find that the mean value of the stopping time is the same for both of these problems. However, for short times, the probability for a walker performing a QW to arrive at the boundary is larger than that for a RW. This means that, although the mean stopping time of a quantum and classical walker are the same, the quantum walker has a greater probability of arriving at the boundary earlier than the classical walker.