Apoorva Patel, K. S. Raghunathan
The spatial search problem on regular lattice structures in $d\geq2$ integer number of dimensions has been studied extensively, using both coined and coinless quantum walks. The Dirac operator has been a crucial ingredient in these studies. In this work we investigate the spatial search problem on fractals of non-integer dimensions. While the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator. We find that the scaling behaviour of 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.
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http://arxiv.org/abs/1203.3950
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