Two-dimensional quantum walk with position-dependent phase defects

Umer Farooq, Abdullah S. Alshammari, Najeh Rekik

Yongzhen Xu, Gongde Guo, Song Lin

Chusei Kiumi, Kei Saito

Rong Zhang, Peng Xue, Jason Twamley

We observe the localization effect of one-dimensional quantum walks with single-point phase defects. The walker's spread velocity is dramatically suppressed by interference effects due to the phase defect. We show that the localization effect depends on four factors: the value and the position of the phase defect, the parameter of coin flipping, and the initial state of the walker + coin system.

Z. J. Li, Josh Izaac, Jingbo B. Wang

We investigate the scattering properties of quantum walks by considering single and double position defects on a one-dimensional line. This corresponds to introducing, at designated positions, delta potential defects for continuous-time quantum walks and phase-defect Hadamard coins for discrete time quantum walks. The delta potential defects can be readily considered as potential barriers in discrete position space, which affect the time evolution of the system in a similar way as the quantum wave-packet dynamics in a continuous position space governed by Schr\"odinger's equation. Although there is no direct analogy of potential barriers in the theoretical formulation of discrete time quantum walks, in this paper we show that the phase defects in the coin space can be utilized to provide similar scattering effects. This study provides means of controlling the scattering properties of quantum walks by introducing designated position-dependent defects.

Shimpei ENDO, Takako Endo, Norio Konno +2 more

We attempt to analyze a one-dimensional space-inhomogeneous quantum walk (QW) with one defect at the origin, which has two different quantum coins in positive and negative parts. We call the QW ``the two-phase QW with one defect'', which we treated concerning localization theorems. The two-phase QW with one defect has been expected to be a mathematical model of topological insulator which is an intense issue both theoretically and experimentally. In this paper, we derive the weak limit theorem describing the ballistic spreading, and as a result, we obtain the mathematical expression of the whole picture of the asymptotic behavior. Our approach is based mainly on the generating function of the weight of the passages. We emphasize that the time-averaged limit measure is symmetric for the origin, however, the weak limit measure is asymmetric, which implies that the weak limit theorem represents the asymmetry of the probability distribution.