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arXiv:2603.05643 · Physical Review A

Localization Without Disorder: Quantum Walks on Structured Graphs

Shyam Dhamapurkar, K. Venkata Subrahmanyam

quant-phSubmitted Mar 5, 2026• Announced Mar 9, 2026• 0 citations
arXivHTMLPDF

Abstract

Continuous-time quantum walks (CTQWs) exhibit localization phenomena that differ fundamentally from their classical counterparts, yet the precise relationship between network structure, spectral degeneracy, and confined dynamics remains incompletely understood. In this work, we present a complete analytical characterization of localization in CTQWs on two highly symmetric graph families: barbell graphs and star-of-cliques graphs. These networks combine pronounced spectral degeneracy with modular structure, enabling exact diagonalization and explicit computation of both eigenstate and dynamical inverse participation ratios (IPRs). Our analysis reveals that localization is governed by the interplay between degenerate subspaces, which generate families of confined modes, and hybridization between invariant subspaces, which redistributes spectral weight. Notably, the dynamical IPR can exceed expectations based solely on eigenstate IPRs, demonstrating that coherent superposition within degenerate eigenspaces enhances confinement. By connecting IPR values to the effective number of vertices visited, we provide a structural diagnostic for predicting quantum transport outcomes in modular networks, establishing that connectivity alone can determine where and how strongly a quantum walk localizes.

Journal reference
Phys. Rev. A 114, 022215 (2026)
DOI
10.1103/bzgp-lpdc
Comments
27 pages, 3 figures

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Continuous-time quantum walks (CTQWs) exhibit localization phenomena that differ fundamentally from their classical counterparts, yet the precise relationship between network structure, spectral degeneracy, and confined dynamics remains incompletely understood. In this work, we present a complete analytical characterization of localization in CTQWs on two highly symmetric graph families: barbell graphs and star-of-cliques graphs. These networks combine pronounced spectral degeneracy with modular structure, enabling exact diagonalization and explicit computation of both eigenstate and dynamical inverse participation ratios (IPRs). Our analysis reveals that localization is governed by the interplay between degenerate subspaces, which generate families of confined modes, and hybridization between invariant subspaces, which redistributes spectral weight. Notably, the dynamical IPR can exceed expectations based solely on eigenstate IPRs, demonstrating that coherent superposition within degenerate eigenspaces enhances confinement. By connecting IPR values to the effective number of vertices visited, we provide a structural diagnostic for predicting quantum transport outcomes in modular networks, establishing that connectivity alone can determine where and how strongly a quantum walk localizes.
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.
We show by general arguments that networks whose density of states contains few highly degenerate eigenvalues result in inefficient performances of continuous-time quantum walks (CTQW) over these networks, while systems whose eigenvalues all have the same degeneracy lead to very efficient transport. We exemplify our results by considering CTQW and, for comparison, its classical counterpart, continuous-time random walks, over simple structures, whose eigenvalues and eigenstates can be calculated analytically. Extensions to more complicated, hyper-branched networks are discussed.
We present a quantum-dynamical framework for identifying structurally important residues in proteins based on continuous time quantum walks (CTQWs) on weighted residue interaction networks constructed from experimentally resolved structures. By mapping the weighted adjacency matrix to a Hamiltonian, residue importance emerges from the long-time averaged occupation probability, confirmed analytically through its spectral decomposition. Across a dataset of approximately 150 proteins spanning diverse structural and functional classes, CTQW centrality exhibits consistently strong agreement with classical eigenvector centrality in identifying central residues, while extending beyond it through incorporating signatures of quantum interference. Analyzing the time-averaged quantum transition matrix reveals consistently larger spectral gaps than the classical random-walk operator. Furthermore, biological relevance is confirmed through recovery of experimentally established functional residues in proteins kinase A and oxytocin. CTQW-derived centrality rankings are accessible on near-term intermediate-scale quantum hardware, as we demonstrate through a proof-of-principle implementation on IBM superconducting quantum hardware. These results establish continuous-time quantum walks as a computationally tractable framework for protein network analysis, that connects network theoretical treatments of protein structural biology to continuous-time quantum walk dynamics.
Continuous-time quantum walks (CTQWs) provide a valuable model for quantum transport, universal quantum computation, and quantum spatial search, among others. Recently, the empowering role of new degrees of freedom in the Hamiltonian generator of CTQWs, which are the complex phases along the loops of the underlying graph, was acknowledged for its interest in optimizing or suppressing transport on specific topologies. We argue that the quantum-classical distance, a figure of merit which was introduced to capture the difference in dynamics between a CTQW and its classical, stochastic counterpart, guides the optimization of parameters of the Hamiltonian to achieve better quantum transport on cycle graphs and spatial search for the quantum speed limit without an oracle on complete graphs, the latter also implying fast uniform mixing. We compare the variations of this quantity with the 1-norm of coherence and the inverse participation ratio, showing that the quantum-classical distance is linked to both, but in a topology-dependent relation, which is key to spot the most interesting quantum evolution in each case.
Continuous time quantum walks provide an important framework for designing new algorithms and modelling quantum transport and state transfer problems. Often, the graph representing the structure of a problem contains certain symmetries that confine the dynamics to a smaller subspace of the full Hilbert space. In this work, we use invariant subspace methods, that can be computed systematically using the Lanczos algorithm, to obtain the reduced set of states that encompass the dynamics of the problem at hand without the specific knowledge of underlying symmetries. First, we apply this method to obtain new instances of graphs where the spatial quantum search algorithm is optimal: complete graphs with broken links and complete bipartite graphs, in particular, the star graph. These examples show that regularity and high-connectivity are not needed to achieve optimal spatial search. We also show that this method considerably simplifies the calculation of quantum transport efficiencies. Furthermore, we observe improved efficiencies by removing a few links from highly symmetric graphs. Finally, we show that this reduction method also allows us to obtain an upper bound for the fidelity of a single qubit transfer on an XY spin network.
We present a quantum-dynamical framework for identifying structurally and functionally important residues in proteins based on continuous-time quantum walks (CTQWs) on weighted residue-interaction networks constructed from experimentally resolved structures. By mapping the weighted adjacency matrix to a Hamiltonian, residue importance emerges from the long-time averaged occupation probability, confirmed analytically through its spectral decomposition. Across a data set of 150 proteins spanning diverse structural and functional classes, CTQW centrality exhibits consistently strong agreement with classical eigenvector centrality in identifying central residues, while extending beyond it through incorporating signatures of quantum interference. Analyzing the time-averaged quantum transition matrix reveals consistently larger spectral gaps than the classical random-walk operator. Furthermore, biological relevance is confirmed through recovery of experimentally established functional residues in proteins kinase A and oxytocin. CTQW-derived centrality rankings are accessible on near-term intermediate-scale quantum hardware, as we demonstrate through a proof-of-principle implementation on IBM superconducting quantum hardware. These results establish continuous-time quantum walks as a computationally tractable framework for protein network analysis, that connects network theoretical treatments of protein structural biology to continuous-time quantum walk dynamics.