The localization of quantum random walks on sierpinski gaskets

TL;DR

This paper investigates quantum random walks on Sierpinski gaskets, deriving recursive formulas for Green functions, and demonstrates their recurrence property as the fractal level increases.

Contribution

It introduces exact recursive formulas for amplitude Green functions on fractals and analyzes the recurrence of quantum walks on Sierpinski gaskets.

Findings

Quantum walks are recurrent on Sierpinski gaskets as the level increases.

Recursive formulas enable precise calculation of hitting probabilities.

Numerical evaluations support theoretical recurrence results.

Abstract

We consider the discrete time quantum random walks on a Sierpinski gasket. We study the hitting probability as the level of fractal goes to infinity in terms of their localization exponents $\beta_w$ , total variation exponents $\delta_w$ and relative entropy exponents $\eta_w$ . 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, based on which the hitting probabilities and expectation of the first-passage time are calculated. 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 origin, i.e., the quantum walks on Sierpinski gasket are recurrent.