LIMIT THEOREMS FOR THE DISCRETE-TIME QUANTUM WALK ON A GRAPH WITH JOINED HALF LINES

We consider a discrete-time quantum walk W-t,W-kappa at time t on a graph with joined half lines J(kappa), which is composed of kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W-t,W-kappa can be reduced to the walk on a half line even if the initial state is asymmetric. For W-t,W-kappa, we obtain two types of limit theorems. The first one is an asymptotic behavior of W-t,W-kappa, which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W-t,W-kappa. On each half line, W-t,W-kappa converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.