Scaling Hypothesis of Spatial Search on Fractal Lattice Using Quantum Walk

We investigate a quantum spatial search problem on a fractal lattice. A recent study for the Sierpinski gasket and tetrahedron made a conjecture that the dynamics of the search on a fractal lattice is determined by spectral dimension for the optimal oracle calls, and not by the fractal dimension [A. Patel and K. S. Raghunathan, Phys. Rev. A 86, 012332 (2012)]. We tackle this problem for the Sierpinski carpet, and we find that our simulation result may support the conjecture. We also propose a scaling hypothesis of oracle calls for the quantum amplitude amplification in a fractal lattice, which is given by the Euclidean dimension, fractal dimension, spectral dimension, and the scale factor of a fractal lattice. We have confirmed that our scaling hypothesis holds in the Sierpinski carpet, gasket, and tetrahedron.