TY - JOUR

T1 - Scaling hypothesis of a spatial search on fractal lattices using a quantum walk

AU - Sato, Rei

AU - Nikuni, Tetsuro

AU - Watabe, Shohei

N1 - Funding Information:
S.W. was supported by JSPS KAKENHI Grant No. JP18K03499. T.N. was supported by JSPS KAKENHI Grant No. JP16K05504. We thank Apoorva Patel for a useful discussion about the scaling hypothesis related to the number of oracle calls. We also thank T. Akimoto for discussions about the classical random walk and the fractal lattice.
Publisher Copyright:
© 2020 American Physical Society.

PY - 2020/2

Y1 - 2020/2

N2 - 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 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 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.

AB - 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 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 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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U2 - 10.1103/PhysRevA.101.022312

DO - 10.1103/PhysRevA.101.022312

M3 - Article

AN - SCOPUS:85079773942

VL - 101

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 2

M1 - 022312

ER -