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.
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.
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.
One of the most important algorithmic applications of quantum walks is to solve spatial search problems. A widely used quantum algorithm for this problem, introduced by Childs and Goldstone [Phys. Rev. A 70, 022314 (2004)], finds a marked node on a graph of n nodes via a continuous-time quantum walk. This algorithm is said to be optimal if it can find any of the nodes in O(n) time. However, given a graph, no general conditions for the optimality of the algorithm are known and previous works demonstrating optimal quantum search for certain graphs required an instance-specific analysis. In fact, the demonstration of the necessary and sufficient conditions that a graph must fulfill for quantum search to be optimal has been a long-standing open problem. In this work we make significant progress towards solving this problem. We derive general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled. Our predictions are valid, for example, for (normalized) Hamiltonians whose spectral gap is considerably larger than ${n}^{\ensuremath{-}1/2}$. This allows us to derive necessary and sufficient conditions for optimal quantum search in this regime, as well as provide examples of graphs where quantum search is suboptimal. In addition, by extending this analysis, we are also able to show the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitioned into clusters. Our results imply that, to the best of our knowledge, all prior results analytically demonstrating the optimality of this algorithm for specific graphs can be recovered from our general results.
Complex networks are structurally disordered systems that often display clustering behavior. The emergent clusters, also known as communities, consist of nodes that are more connected among themselves than they are connected with the rest of the network. Analyzing community structure is an important problem in network theory, with numerous applications in different fields. In this work I investigate the evolution of a continuoustime quantum random walk on a social network with benchmark community structure and show that it can be used to perform community detection.
We introduce a geometric framework for continuous-time quantum walks on graphs by embedding each vertex into a Euclidean space through its time-dependent quantum probability distribution. This construction induces a rich geometry in which quantum transport is characterized by distances, radii, angles, and simplex volumes, allowing interference, localization, and spreading to be analyzed within a unified metric-angular formalism. We prove that, in contrast to classical diffusion, which collapses to a spherical geometry, quantum dynamics generate a generically non-spherical affine geometry with persistent anisotropy. Applying this theory to real-world networks—including transportation systems, semantic graphs, and neuronal connectomes—we show that quantum geometry reveals dynamically meaningful backbones, interference-based “communities”, and vulnerability structures that are invisible to classical random-walk and spectral methods. In particular, angular and radial quantum descriptors isolate functional hubs, control cores, and coherence classes without any topological or dimensionality assumptions. Together, these results demonstrate that quantum-walk-induced geometry provides a powerful new lens for understanding structure and function in complex networks.
Abstract Understanding how decoherence disrupts quantum walks is vital for developing robust quantum technologies, as environmental interactions often degrade quantum properties or induce transitions to classical random walks. We investigate the stability of continuous-time quantum walks across different network topologies subjected to three distinct decoherence mechanisms: energy-based intrinsic decoherence, node-based Haken–Strobl noise, and edge-based quantum stochastic walk. By defining stability as the preservation of quantum properties, we find that densely connected networks, such as the complete network, and hub dominated networks, such as scale-free and star topologies, are stable under Haken–Strobl noise but become uniquely vulnerable under quantum stochastic walk. However, these same networks exhibit lower coherence in the noiseless regime due to their inherent localization, highlighting a fundamental trade-off between localization and coherence. Furthermore, the centrality of the initialization node has a pronounced impact on relaxation time, underscoring the critical role of local topological features in quantum dynamics.
Localization of quantum walks can be characterized by the return probability, i.e., the probability for the walker returning to its original site. In this paper, we consider localization of continuous-time quantum walks in terms of return probability on a deterministic recursive tree, which is generated by adding one node and connecting it to each node of the existing tree recursively. We obtain an approximate form for the return probability using the complete eigenvalues and eigenstates of Laplace matrix of the structure. It is found that the return probability depends on the initial node of the excitation. When the walk starts at the central nodes, the return probability converges to a constant value even in the limit of infinite system, in contrast to an exponential decay of the return probability if the walk starts at the outlying nodes. We also observe a bipartite structure for the distribution of return probability, and provide theoretical interpretation for all our findings. Our results suggest that quantum walks display significant localization and well-bedded structure of return probability on heterogeneous trees.