Spatial search by continuous-time quantum walks on renormalized Internet networks

Abstract

We study spatial search with continuous-time quantum walks on real complex networks. We use smaller replicas of the Internet network obtained with a recent geometric renormalization method introduced by García- Pérez et al. (2018). This allows us to infer for the first time the behavior of a quantum spatial search algorithm on a real complex network. By simulating numerically the dynamics and optimizing the coupling parameter, we study the optimality of the algorithm and its scaling with the size of the network, showing that on average it is considerably better than the classical scaling O(N ), but it does not reach the ideal quadratic speedup O( √ N ) that can be achieved, e.g. in complete graphs. However, the performance of the search algorithm strongly depends on the degree of the nodes and, in fact, the scaling is found to be very close to optimal when we consider the nodes below the 90th percentile ordered according to the degree.

Ion trap long-range XY model for quantum state transfer and optimal spatial search

“…Childs and Goldstone [13] found that the spatial search problem for the complete graph, the hypercube graph, and d-dimensional periodic lattices of d > 4 can be solved in optimal time using continuous-time quantum walks. A number of high dimensional graphs were subsequently found that permit optimal spatial search [14][15][16][17][18][19][20][21][22][23]. Recently, optimal spatial search in one dimension using long-range interactions was found to be possible [24].…”