Search on a Fractal Lattice using a Quantum Random Walk

TL;DR

This paper explores quantum spatial search algorithms on fractal structures, specifically Sierpinski gaskets, revealing that the search efficiency depends on spectral dimension rather than fractal dimension.

Contribution

It introduces a quantum walk model on fractals using a Klein-Gordon mode, extending quantum search analysis beyond regular lattices.

Findings

Search scaling determined by spectral dimension

Quantum walk constructed without Dirac operator

Numerical results on Sierpinski gaskets in 2D and 3D

Abstract

The spatial search problem on regular lattice structures in integer number of dimensions $d\geq2$ has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here we investigate the spatial search problem on fractals of non-integer dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.