Continuous Time Quantum Walks in finite Dimensions

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.