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.
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.
Quantum geometry is conventionally understood as a descriptive structure characterizing the space of quantum states. In this work, we examine an alternative but fully compatible interpretation in which geometric quantities—such as Berry curvature and the quantum metric—function effectively as constraint layers shaping the set of admissible dynamical trajectories. Without introducing new degrees of freedom or modifying established theoretical frameworks, we analyze how geometric terms enter semiclassical equations of motion and transport responses in quantum systems. In particular, anomalous velocity contributions and geometry-dependent transport coefficients indicate that geometry operates not merely as a passive descriptor, but as an active structural condition on system evolution. This shift from geometry as description to geometry as constraint does not alter the formalism, but reorganizes its interpretation, reducing conceptual fragmentation by unifying diverse geometry-dependent effects under a common structural principle. It highlights that the evolution of quantum systems is not only governed by dynamical variables, but also by the geometric structure that restricts the space of accessible trajectories. This perspective suggests that transport phenomena may be more fruitfully analyzed through constraint-space formulations rather than purely state-space descriptions. We clarify the scope and limits of this reinterpretation, emphasizing that it introduces no new physical interactions or ontological commitments. Instead, it provides a conceptually sharper and operationally informative framework for understanding the role of geometry in quantum dynamics and transport.
Quantum communication networks—the substrate of a future quantum Internet—demand analytical tools that account for entanglement, fidelity, and quantum-specific constraints absent from classical models. In this paper, we introduce the Quantum Network Science (QNS) framework that adapts core network metrics to the quantum setting through fidelity-, success-probability-, and capacity-aware weighting. We formalize centrality (including Quantum PageRank and continuous-time quantum-walk variants), community structure on entanglement graphs, and robustness/percolation with fidelity thresholds. The framework is validated via analytic motifs and controlled simulations on Erdős–Rényi, scale-free, and small-world topologies, as well as satellite-assisted versus fiber-only designs. Our results show that (i) fidelity weighting reorders structural importance and can reconnect networks that appear fragmented classically; (ii) heavy-tailed degree patterns improve tolerance to random failures but heighten vulnerability to targeted hub attacks; (iii) small-world shortcuts induced by long-range quantum links shrink path lengths; and (iv) overlapping “connected components” emerge from entanglement swapping, motivating revised connectivity baselines. We also discuss design implications—degree caps and hub hardening, link-type diversity, multipath routing, and buffering policies—and outline extensions to temporal and multilayer modeling that couple the quantum plane with its classical control layer. QNS thus offers a principled, measurement-oriented foundation for analyzing, comparing, and engineering resilient, high-capacity quantum networks.
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.
This paper explores the potential of non-Euclidean geometry to provide a novel framework for understanding and modeling quantum information. Traditional approaches often treat quantum information as a purely mathematical construct, limiting its applicability to purely computational contexts. We propose a radical shift, formalizing quantum information as a geometric object within a non-Euclidean space, leveraging the properties of geometry to reveal deeper connections to the physical world. The core mechanism centers around extending the concept of quantum information beyond classical representations, allowing for the visualization and manipulation of entanglement and other quantum phenomena through a geometric lens. This paper will detail the conceptual groundwork, mathematical framework, and potential implications of this approach, emphasizing its capacity to bridge the gap between abstract mathematics and the complexities of quantum mechanics. Chapter 1: Foundations – Geometry and Information The fundamental concept underpinning this research lies in the recognition that geometry, particularly non-Euclidean geometry, possesses inherent properties that resonate with the nature of quantum information. Classical geometry, with its inherent Euclidean assumptions, struggles to adequately capture the counterintuitive aspects of quantum mechanics. The inherent non-Euclidean nature of space, where distances are not absolute but relative, mirrors the fundamental fuzziness and probabilistic nature of quantum states. Furthermore, the distribution of information, a cornerstone of quantum mechanics, can be visualized as a complex geometric pattern, and non-Euclidean geometry offers a powerful tool to analyze and manipulate this pattern. The challenge, therefore, is to move beyond a purely mathematical abstraction to a geometric understanding of information itself. Early attempts to model quantum information often relied on representations like Hilbert spaces, which, while powerful, represent information as a vector space. However, this representation lacks the inherent geometric richness that is crucial for understanding entanglement and other quantum phenomena. We propose a shift towards representing quantum information as points within a non-Euclidean space, where the distance and orientation of these points encode the quantum state. This shift is predicated on the observation that geometric properties, such as curvature and connectivity, are intimately linked to quantum entanglement and the preservation of information. The core idea is to formulate a geometric model where entanglement is represented not as a correlation between particles but as a distortion of the geometric space itself, influenced by the quantum state. This distortion, driven by the inherent properties of the non-Euclidean space, provides a richer and potentially more intuitive understanding of quantum phenomena. The choice of non-Euclidean geometry is not arbitrary; it is intended to be a 'natural' choice, reflecting the underlying structure of quantum information. Chapter 2: Non-Euclidean Geometry – A Conceptual Framework We introduce a specific type of non-Euclidean geometry, termed "Hyperbolic Geometry for Quantum Information," that incorporates several key features. Unlike Euclidean geometry, Hyperbolic Geometry allows for the representation of multiple, potentially conflicting, states simultaneously. This is achieved through the concept of "Hyperbolic Density," which quantifies the probability of finding a quantum state within a given region of the space. The density of a Hyperbolic Density is influenced by the quantum state, creating a dynamic and evolving geometric representation. The mathematical framework of Hyperbolic Geometry is built around the concept of "Geometric Curvature." This curvature is not simply a measure of distance but rather a representation of the quantum state's inherent orientation. Specifically, we define a curvature function that is sensitive to the quantum state. This curvature is not constant; it adapts to the quantum state, creating a continuously evolving geometric representation. This adaptation is crucial for capturing the inherent randomness and non-determinism of quantum mechanics. Furthermore, we introduce the concept of "Geometric Connectivity," which describes the relationships between points within the Hyperbolic Geometry. These connections are not simply spatial relationships but are defined by the quantum state's inherent properties. The strength and type of connectivity represent the degree to which the quantum state is entangled with the geometric structure. This connectivity is dynamically modified by the curvature function, creating a complex and interconnected geometric pattern. The core mathematical tool for manipulating these geometries is the "Geometric Transformation." This transformation allows us to 'morph' geometric forms into new forms, representing the evolution of quantum information. The transformation is based on a specific set of rules which are intrinsically linked to the nature of entanglement. These rules are carefully designed to preserve the fundamental properties of quantum information.
Quantum walks provide a natural framework to approach graph problems with quantum computers, exhibiting speedups over their classical counterparts for tasks such as the search for marked nodes or the prediction of missing links. Continuous-time quantum walk algorithms assume that we can simulate the dynamics of quantum systems where the Hamiltonian is given by the adjacency matrix of the graph. It is known that such can be simulated efficiently if the underlying graph is row-sparse and efficiently row-computable. While this is sufficient for many applications, it limits the applicability for this class of algorithms to study real world complex networks, which, among other properties, are characterized by the existence of a few densely connected nodes, called hubs. In other words, complex networks are typically not row-sparse, even though the average connectivity over all nodes can be very small. In this work, we extend the state-of-the-art results on quantum simulation to graphs that contain a small number of hubs, but that are otherwise sparse. Hopefully, our results may lead to new applications of quantum computing to network science.