Eur. Phys. J. D (2012) 66: 142
https://doi.org/10.1140/epjd/e2012-30146-9

Regular Article

Continuous deformations of the Grover walk preserving localization

Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Praha 1 – Staré Město, Czech Republic

^a e-mail: martin.stefanak@fjfi.cvut.cz

Received: 2 March 2012
Received in final form: 30 March 2012
Published online: 23 May 2012

Abstract

The three-state Grover walk on a line exhibits the localization effect characterized by a non-vanishing probability of the particle to stay at the origin. We present two continuous deformations of the Grover walk which preserve its localization nature. The resulting quantum walks differ in the rate at which they spread through the lattice. The velocities of the left and right-traveling probability peaks are given by the maximum of the group velocity. We find the explicit form of peak velocities in dependence on the coin parameter. Our results show that localization of the quantum walk is not a singular property of an isolated coin operator but can be found for entire families of coins.