Papers › Scaling Hypothesis of Spatial Search on Fractal Lattice Using Quantum Walk
Scaling Hypothesis of Spatial Search on Fractal Lattice Using Quantum Walk
Rei Sato, Tetsuro Nikuni, Shohei Watabe
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We investigate a quantum spatial search problem on fractal lattices, such as Sierpinski carpets and Menger sponges. In earlier numerical studies of the Sierpinski gasket, the Sierpinski tetrahedron, and the Sierpinski carpet, conjectures have been proposed for the scaling of a quantum spatial search problem finding a specific target, which is given in terms of the characteristic quantities of a fractal geometry. We find that our simulation results for extended Sierpinski carpets and Menger sponges support the conjecture for the optimal number of the oracle calls, where the exponent is given by 1/2 for dₛ > 2 and the inverse of the spectral dimension dₛ for dₛ < 2. We also propose a scaling hypothesis for the effective number of the oracle calls defined by the ratio of the optimal number of oracle calls to a square root of the maximum finding probability. The form of the scaling hypothesis for extended Sierpinski carpets is very similar but slightly different from the earlier conjecture for the Sierpinski gasket, the Sierpinski tetrahedron, and the conventional Sierpinski carpet.
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