The localization of quantum random walks on sierpinski gaskets

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.

References in corpus (8)

• Quantum walks: a comprehensive review
• Spanning trees on the Sierpinski gasket
• Search on a Fractal Lattice using a Quantum Random Walk
• The Limiting Distribution of Decoherent Quantum Random Walks
• Open quantum random walks with decoherence on coins with $n$ degrees of freedom
• Convergence of quantum random walks with decoherence
• Coins Make Quantum Walks Faster
• A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpinski gasket