Clustering-induced localization of quantum walks on networks

TL;DR

This paper investigates how local clustering in networks induces localization of quantum walks, providing analytical expressions and demonstrating this effect across various network models.

Contribution

It reveals the role of network clustering in quantum walk localization and derives an analytical formula for long-time behavior based on eigenvector products.

Findings

Localization occurs in highly clustered networks.

Analytical expression for inverse participation ratio derived.

Localization also observed in small-world and power-law cluster 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 the localization of quantum walks on networks. We demonstrate how localization emerges in highly clustered networks that we construct by recursively attaching triangles, and we derive an analytical expression for the long-time inverse participation ratio that depends on products of eigenvectors of the quantum-walk Hamiltonian. 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.