Discrete time quantum walk on the Apollonian network

This work discusses the discrete time quantum walk (DTQW) on the first five generations of the Apollonian network. This structure is constructed in a geometric recurrent way and is characterized by, among other features, a scale-free distribution of node degrees. The DTQW formalism requires a node-dependent coin operator that, for each node, has as many different output states as that node degree, so that each local coin operator is expressed by the Fourier operator. The DTQW time evolution matrix has a larger rank than that of the network adjacency matrix or the matrix representation of the continuous time quantum walk (CTQW). Results for the time evolution, return time to specific nodes and asymptotic probability of site occupancy are obtained. Such DTQW-specific features are discussed and compared to those obtained for the classical random walk and CTQW on the same lattice. (paper)