Caminhadas quânticas em tempo discreto em rede pequeno mundo

Abstract

The quantum walks, besides being naturally used in the modeling of systems of this scale, has been widely used in the building quantum algorithms. In the last, both its formulation in continuous and discrete time, in some cases, presents better performance when compared to algorithms that use classical random walk. Among these formulations, the literature has highlighted the use of quantum walk in discrete time, in particular, its applicability in complex networks. The small world network is part of this type of structure. In this work, the discrete time quantum walk on this network is presented, analyzing the behavior of the probability distribution with the variation of the parameters q, which determines the topology of the network, and k, which defines the neighborhood closest to the vertex. A formalism is defined that describes this dynamic, allowing to numerically simulate the walk. The evolution of the probability distributions versus parameters reveals its tendency to spread, towards uniformity. Between this transition, there is a regime different from the characteristic ballistic behavior in a circular network. There is an attempt to determine a law of scale to describe this evolution, working only for a small range of parameters.