Quantum Walks on Sierpinski Gaskets

TL;DR

This paper investigates quantum walks on Sierpinski gaskets, revealing how fractal structure influences quantum diffusion and mixing times, with results differing from regular lattices and showing classical-like behavior on average.

Contribution

It provides the first detailed analysis of quantum walks on fractal structures, specifically Sierpinski gaskets, highlighting the impact of fractal geometry on quantum diffusion properties.

Findings

Displacement depends on initial location.

Average displacement exponent similar to classical diffusion.

Quantum walk behavior differs from regular lattices.

Abstract

We analyze discrete-time quantum walks on Sierpinski gaskets using a flip-flop shift operator with the Grover coin. We obtain the scaling of two important physical quantities: the mean-square displacement and the mixing time as function of the number of points. The Sierpinski gasket is a fractal that lacks translational invariance and the results differ from those described in the literature for ordinary lattices. We find that the displacement varies with the initial location. Averaged over all initial locations, our simulation obtain an exponent very similar to classical diffusion.