Clustering-induced localization of quantum walks on networks.

Abstract

Quantum walks on networks are a paradigmatic model in quantum information theory. Quantum-walk algorithms have been developed for various applications, including spatial-search problems, element-distinctness problems, and node-centrality analysis. Unlike their classical counterparts, the evolution of quantum walks is unitary, so they do not converge to a stationary distribution. However, for many applications, it is important to understand the long-time behavior of quantum walks and the impact of network structure on their evolution. In the present paper, we study localization of continuous-time quantum walks on networks. We derive an analytical expression for the long-time inverse participation ratio that depends on combinations of eigenvectors of the quantum-walk Hamiltonian. We then demonstrate how localization emerges in highly clustered networks that we construct by recursively attaching triangles. Building on the insights from this example, we then show that localization also occurs in Kleinberg navigable small-world networks and Holme-Kim power-law cluster networks. Our results illustrate that local clustering, which is a key structural feature of networks, can induce localization of quantum walks.