Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory

We analyze continuous-time quantum and classical random walk on spidernet lattices. In the framework of Stieltjes transform, we obtain density of states, which is an efficiency measure for the performance of classical and quantum mechanical transport processes on graphs, and calculate the spacetime transition probabilities between two vertices of the lattice. Then we analytically show that there are two power law decays $\sim t^{-3}$ and $\sim t^{-1.5}$ at the beginning of the transport for transition probability in the continuous-time quantum and classical random walk respectively. This results illustrate the decay of quantum mechanical transport processes is quicker than that of the classical one. Due to the result, the characteristic time $t_c$, which is the time when the first maximum of the probabilities occur on an infinite graph, for the quantum walk is shorter than that of the classical walk. Therefore, we can interpret that the quantum transport speed on spidernet is faster than that of the classical one. In the end, we investigate the results by numerical analysis for two examples.