Quantum spatial search on planar networks

Abstract: This paper deals with the problem of the requirements for quantum systems that enable one to design efficient quantum algorithms. We rise the issue of the possibility to utilise the non-complete networks for algorithmic purposes. In particular we consider applications for the spatial search problems. We focus on showing that the asymptotic complexity widely discussed in the related work is not enough tool for determining the potential of the network. We provide an example of a network where the asymptotic complexity is the same for a variety of cases and yet it is not always possible to implement successful search procedure within the quantum walk scheme. The examples are based on an Apollonian network which models a variety of iteratively generated planar networks. The network is planar, exhibits linear growth of edges number, consists nodes of different degrees and has the small-world and scale-free properties. This motivates its analysis due to the simplicity in terms of connections density and potential for quantum phenomena due to the nodes diversity.