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

TL;DR

This study investigates the performance of continuous-time quantum walks for spatial search on real-world Internet networks, revealing near-optimal scaling for low-degree nodes but not achieving the ideal quadratic speedup.

Contribution

First application of quantum spatial search algorithms to real-world complex networks using geometric renormalization techniques.

Findings

Quantum search scales better than classical $\\mathcal{O}(N)$ but not as well as ideal $\\mathcal{O}(\sqrt{N})$.

Performance depends strongly on node degree, with near-optimal results for nodes below the 99th percentile.

Scaling is close to optimal for low-degree nodes, highlighting the importance of network topology.

Abstract

We study spatial search with continuous-time quantum walks on real-world complex networks. We use smaller replicas of the Internet network obtained with a recent geometric renormalization method introduced by Garc\'ia-P\'erez et al., Nat. Phys. 14, 583 (2018). This allows us to infer for the first time the behavior of a quantum spatial search algorithm on a real-world 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 $\mathcal{O}(N)$, but it does not reach the ideal quadratic speedup $\mathcal{O}(\sqrt{N})$ 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 $99$th percentile ordered according to the degree.