Continuous Time Quantum Walks in finite Dimensions

TL;DR

This paper investigates how the spectral dimension of networks influences the efficiency of continuous time quantum walks in quantum search problems, revealing that optimal search occurs when the spectral dimension exceeds four.

Contribution

It introduces the role of spectral dimension in quantum search complexity on fractal and general networks, extending previous lattice results to fractals.

Findings

Quantum search complexity scales with spectral dimension d_s.

Optimal quantum search (Grover limit) occurs when d_s > 4.

Results generalize previous lattice-based findings to fractal networks.

Abstract

We consider the quantum search problem with a continuous time quantum walk for networks of finite spectral dimension d_{s} of the network Laplacian. For general networks of fractal (integer or non-integer) dimension d_{f}, for which in general d_{f}\not=d_{s}, it suggests that d_{s} is the scaling exponent that determines the computational complexity of the search. Our results are consistent with those of Childs and Goldstone [Phys. Rev. A 70 (2004), 022314] for lattices of integer dimension, where d=d_{f}=d_{s}. For general fractals, we find 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}\to\infty) networks by Chakraborty et al. [Phys. Rev. Lett. 116 (2016), 100501] showing that for all those networks spatial search by quantum walk is optimal.