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Quantum Information Processing

Quantum walk with quadratic position-dependent phase defects

Umer Farooq, Abdullah S. Alshammari, Najeh Rekik

Published Nov 26, 2019• 3 citations
Publisher
DOI
10.1007/s11128-019-2496-y

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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.
We study the localization of probability distribution in a discrete quantum random walk on an infinite chain. With a phase defect introduced in any position of the quantum random walk (QRW), we have found that the localization of the probability distribution in the QRW emerges. Different localized behaviors of the probability distribution in the QRW are presented when the defect occupies different positions. Given that the coefficients of the localized stationary eigenstates relies on the coin operator, we reveal that when the defect occupies different positions, the amplitude of localized probability distribution in the QRW exhibits a non-trivial dependence on the coin operator.