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Clustering-induced localization of quantum walks on networks
Lucas Böttcher1,2, Mason A Porter3,4,5
1Frankfurt School of Finance and Management, Department of Computational Science and Philosophy, 60322 Frankfurt am Main, Germany.
Physical Review. E
|January 21, 2026
Summary
Quantum walks on networks exhibit localization, a phenomenon influenced by network structure. This study reveals that local clustering in networks can induce this localization, impacting quantum information processing.
Area of Science:
- Quantum Information Theory
- Network Science
- Condensed Matter Physics
Background:
- Quantum walks are essential models in quantum information theory with applications in search and analysis.
- Unlike classical walks, quantum walks evolve unitarily and do not reach a stationary distribution.
- Understanding the long-time behavior and network impact on quantum walks is crucial for applications.
Purpose of the Study:
- To investigate the phenomenon of localization in continuous-time quantum walks on networks.
- To derive an analytical expression for the long-time inverse participation ratio.
- To explore the relationship between network structure, particularly local clustering, and quantum walk localization.
Main Methods:
- Derivation of an analytical expression for the long-time inverse participation ratio.
- Analysis of quantum walks on specifically constructed highly clustered networks (recursively attached triangles).
- Examination of quantum walks on established network models: Kleinberg navigable small-world and Holme-Kim power-law cluster networks.
Main Results:
- An analytical expression for the long-time inverse participation ratio was derived, dependent on eigenvectors of the quantum-walk Hamiltonian.
- Localization was observed in constructed highly clustered networks.
- Localization was also demonstrated in Kleinberg navigable small-world and Holme-Kim power-law cluster networks, confirming the role of local clustering.
Conclusions:
- Local clustering is a key network feature that can induce localization of continuous-time quantum walks.
- The findings provide insights into the long-time dynamics of quantum walks and their dependence on network topology.
- This research contributes to understanding quantum walk behavior for potential applications in quantum information processing and network analysis.
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