Asymmetries in symmetric quantum walks on two-dimensional networks

Abstract

We study numerically the behavior of continuous-time quantum walks over networks which are topologically equivalent to square lattices. On short time scales, when placing the initial excitation at a corner of the network, we observe a fast, directed transport through the network to the opposite corner. This transport is not ballistic in nature, but rather produced by quantum mechanical interference. In the long time limit, certain walks show an asymmetric limiting probability distribution; this feature depends on the starting site and, remarkably, on the precise size of the network. The limiting probability distributions show patterns which are correlated with the initial condition. This might have consequences for the application of continuous time quantum walk algorithms.Comment: 9 pages, 12 figures, revtex

Limit distributions of two-dimensional quantum walks

“…It is also an interesting problem to relate the present results to solutions of the continuoustime quantum-walk models on two-dimensional lattices [22].…”

“…One of the recent topics of quantum walks is systematic study of higher dimensional models [14,18,19,20,21,22]. Among them the Grover-walk model has been extensively studied, since it is related to Grover's search algorithm [23,24,25,26,27].…”

Continuous-time quantum walks on Erdös–Rényi networks

“…In the next section, we briefly review the classical and quantum transport on networks presented in Refs. [23,24]. In Section 3 we study the time evolution of the ensemble averaged return probability on ER networks with different parameters.…”

Dynamics of continuous-time quantum walks in restricted geometries

“…Otherwise stated, the Laplacian operator can work both as a classical transfer operator and as a tight-binding Hamiltonian of a quantum transport process [24,25].…”