Scaling Hypothesis of Spatial Search on Fractal Lattice Using Quantum Walk

TL;DR

This study explores quantum spatial search efficiency on fractal lattices, confirming conjectured scaling laws and proposing a new scaling hypothesis for the effective number of oracle calls based on fractal and spectral dimensions.

Contribution

It extends previous conjectures to larger fractal structures and introduces a new scaling hypothesis for the effective oracle calls in quantum search on fractals.

Findings

Optimal oracle call scaling exponent is 1/2 for spectral dimension > 2.

Optimal oracle call scaling is inverse of spectral dimension for spectral dimension < 2.

Proposed a new scaling hypothesis for the effective number of oracle calls.

Abstract

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 ${\it optimal}$ number of the oracle calls, where the exponent is given by $1/2$ for $d_{\rm s} > 2$ and the inverse of the spectral dimension $d_{\rm s}$ for $d_{\rm s} < 2$. We also propose a scaling hypothesis for the ${\it effective}$ number of the oracle calls defined by the ratio of the ${\it optimal}$ 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.