Eur. Phys. J. Plus (2014) 129: 103
https://doi.org/10.1140/epjp/i2014-14103-6

Regular Article

Discrete time quantum walks on percolation graphs

¹ Wigner RCP, SZFKI, Konkoly-Thege M. u. 29-33, H-1121, Budapest, Hungary
² Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, Praha 1 - Staré Město, 115 19, Czech Republic

^* e-mail: kollar.balint@wigner.mta.hu

Received: 12 April 2014
Accepted: 17 April 2014
Published online: 30 May 2014

Abstract

Randomly breaking connections in a graph alters its transport properties, a model used to describe percolation. In the case of quantum walks, dynamic percolation graphs represent a special type of imperfections, where the connections appear and disappear randomly in each step during the time evolution. The resulting open system dynamics is hard to treat numerically in general. We shortly review the literature on this problem. We then present our method to solve the evolution on finite percolation graphs in the long time limit, applying the asymptotic methods concerning random unitary maps. We work out the case of one-dimensional chains in detail and provide a concrete, step-by-step numerical example in order to give more insight into the possible asymptotic behavior. The results about the case of the two-dimensional integer lattice are summarized, focusing on the Grover-type coin operator.