Network topologies for perfect state transfer

For the purpose of processing quantum information, it is important to transfer quantum states from a system to another system with perfect fidelity (i.e., without information loss). When the total system is a spin network, or, equivalently, a noiseless quantum channel, state transfer depends on the arrangement of the subsystems and their couplings. When the couplings are taken to be constant, state transfer depends on the eigensystem of the graph modeling the network, and it requires periodic dynamics. I will review known results about perfect state transfer. Then I will present some new results for a restricted family of graphs, which are usually important topologies for interconnection networks of parallel computers and distributed systems. Specifically, I will give a necessary and sufficient condition on the spectrum of a circulant graph, for modeling a network with periodic dynamics. On the light of this result, I will characterize diameter and order of circulant graphs with integer eingenvalues. Part of this talk is based on work done with Nitin Saxena and Igor Shparlinski. Determine necessary and sufficient conditions on the spectrum of a graph allowing perfect state transfer is an open problem.