Renormalization group for a continuous-time quantum search in finite dimensions
Shanshan Li, Stefan Boettcher · Physical Review A · 2017
We consider the quantum search problem with a continuous-time quantum walk for networks characterized by a finite spectral dimension ${d}_{s}$ of the network Laplacian. For general networks of fractal (integer or noninteger) dimension ${d}_{f}$, for which in general ${d}_{f}\ensuremath{ e}{d}_{s}$, it suggests that it is ${d}_{s}$ that determines the computational complexity of the quantum search. Our results continue those of A. M. Childs and J. Goldstone [Phys. Rev. A 70, 022314 (2004)] for lattices of integer dimension, where $d={d}_{f}={d}_{s}$. Thus, we find for general fractals that the Grover limit of quantum search can be obtained whenever ${d}_{s}>4$. This complements the recent discussion of mean-field (i.e., ${d}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$) networks by S. Chakraborty et al. [Phys. Rev. Lett. 116, 100501 (2016)] showing that for all those networks, spatial search by quantum walk is optimal.