The localization of quantum random walks on Sierpinski gaskets

Abstract

We consider discrete time quantum random walks on a Sierpinski gasket. We study the hitting probability as the level of the fractal graph goes to infinity in terms of their localization exponents βw, total variation exponents δw, and relative entropy exponents ηw. We define and solve the amplitude Green functions recursively when the level of the fractal graph goes to infinity. We obtain exact recursive formulas for the amplitude Green functions, on which the hitting probabilities and expectation of the first-passage time are calculated, and using the recursive formula with the aid of Monte Carlo integration, we evaluate their numerical values. We also show that when the level of the fractal graph goes to infinity, with probability 1, the quantum random walks will return to the origin, i.e., the quantum walks on a Sierpinski gasket are recurrent.