Search on a fractal lattice using a quantum random walk

Abstract

The spatial search problem on regular lattice structures in integer number of dimensions d ≥ 2 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.

Transport properties of continuous-time quantum walks on Sierpinski fractals

“…(10), for successive generations g from 2 to 8, calculated according to Eqs. (19) and (20). Also the exact value of χ lb , see Eq.…”

“…Search via quantum walks on fractal graphs has been considered in Refs. [6,19,20]. A similar mathematical model arises for condensed matter systems, in which one considers a particle moving on an underlying fractal lattice (a Sierpinski gasket); here the solution of Schrödinger's equation has been studied within the tightbinding approximation [21,22].…”

Spatial search algorithms on Hanoi networks

“…[8]. Recently, the abstract search algorithm was applied for spatial search on Sierpinski gasket [9].…”

Quantum Walks on Sierpinski Gaskets

“…The same is true for the Grover coin G. This coin and the°i p-°op shift operator play an important role in spatial search algorithms. 16,13 …”

“…[6][7][8][9] Quantum walks have been analyzed on two-dimensional square lattices, 10,11 and on the hypercube. 12 A spatial search using the discrete-time quantum walk model has been undertaken on the Sierpinski gasket, 13 and on the Hanoi network of degree 3. 14 A quantum walk on the dual Sierpinski gasket using the continuous-time quantum walk model has been analyzed by Agliari et al 15 In this paper, we focus our attention on discrete-time quantum walks on the Sierpinski gasket.…”