Stopping time of a one-dimensional bounded quantum walk

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.

Optical scheme to demonstrate state-independent quantum contextuality

“…In the optical network, [43][44][45][46][47][48][49][50][51][52][53] the first HWP before a PBS can realize the U M in the right-hand side of Fig. 4(a), which converts after the eigenstates to H or V .…”

A two-dimensional quantum walk driven by a single two-side coin*

“…[17,18] The simplest one-dimensional discrete-time QW consists of a single walker, usually being a quantum particle, who takes a step towards left or right on a line according to the outcome of flipping a two-side quantum coin, usually being the internal state of the quantum particle. [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] A higher-dimensional QW has also been proposed and investigated. [34][35][36][37][38][39][40][41] With the number of dimensions increasing, QWs with higher-dimensional walker and multiple-side coin exhibit richer properties [42][43][44][45] and can be used in quantum simulations of many-body systems.…”