Continuous Time Quantum Walk On Complex Networks

Dynamics on complex networks is a hot topic in recent years. Quantum walk, which is a natural extension of the classical random walk, has important applications in the fields of quantum physics, biophysics and quantum informa-tion science. This paper investigates the dynamics of continuous-time quantum walks on complex networks, focusing on two-dimensional networks and scale-free networks. We compare the dynamical behavior between quantum walk and clas-sical random walk, and discuss the impact of network structure on the quantum dynamics.Firstly, we present an extensive analytical study of continuous-time quan-tum walks occurring on two-dimensional lattices under various boundary condi-tions, focusing on the Moobius strips and Klein bottles, which are featured by the twisted boundary conditions. We compare eigenvalue spectra, classical trans-port efficiency, time evolutions of transition probabilities, limiting probabilities for these two graphs and compare them to those for rectangles, cylinders and tori. Our work uncover the effect of various boundary conditions on continuous-time quantum walks occurring on lattices.Secondly, we study continuous-time quantum walks on a class of scale-free net-works controlled by a tunable parameter g, which have the same degree sequence independent of q, but are very different in other structural properties determined by q. We calculate several classical quantities for them, including time evolutions of transition probabilities, transition probabilities between node groups, limiting probabilities, degree-averaged probabilities and so on. Our results suggest that it is far from enough to characterize the behaviors of continuous-time quantum walks on scale-free networks merely by power-law degree distribution.Finally, our network models, intermediate results and analytical technique used here are capable of being extended to many other deterministic networks.