Quantum networks (QNs) exhibit stronger connectivity than predicted by classical percolation, yet the origin of this phenomenon remains unexplored. We apply a statistical physics model—concurrence percolation—to uncover the origin of stronger connectivity on hierarchical scale-free networks, the ( U , V ) flowers. These networks allow full analytical control over path connectivity through two adjustable path-length parameters, ≤ V . This precise control enables us to determine critical exponents well beyond current simulation limits, revealing that classical and concurrence percolations, while both satisfying the hyperscaling relation, fall into distinct universality classes. This distinction arises from how they “superpose” parallel, nonshortest path contributions into overall connectivity. Concurrence percolation, unlike its classical counterpart, is sensitive to nonshortest paths and shows higher resilience to detours as these paths lengthen. This enhanced resilience is also observed in real-world hierarchical, scale-free internet networks. Our findings highlight a crucial principle for QN design: When nonshortest paths are abundant, they notably enhance QN connectivity beyond what is achievable with classical percolation.
Quantum networks (QNs) exhibit stronger connectivity than predicted by classical percolation, yet the origin of this phenomenon remains unexplored. We apply a statistical physics model—concurrence percolation—to uncover the origin of stronger connectivity on hierarchical scale-free networks, the ( U , V ) flowers. These networks allow full analytical control over path connectivity through two adjustable path-length parameters, ≤ V . This precise control enables us to determine critical exponents well beyond current simulation limits, revealing that classical and concurrence percolations, while both satisfying the hyperscaling relation, fall into distinct universality classes. This distinction arises from how they “superpose” parallel, nonshortest path contributions into overall connectivity. Concurrence percolation, unlike its classical counterpart, is sensitive to nonshortest paths and shows higher resilience to detours as these paths lengthen. This enhanced resilience is also observed in real-world hierarchical, scale-free internet networks. Our findings highlight a crucial principle for QN design: When nonshortest paths are abundant, they notably enhance QN connectivity beyond what is achievable with classical percolation.
The evolution of connectivity in quantum networks under decoherence and link degradation is a central problem in quantum information, calling for further understanding of the nature of its transition during structural network degradation. By diluting each link with probability 1−f, we focus on connectivity strength transitions in diluted hierarchical scale-free quantum networks, the (u,v) flowers, which are analytically tractable through two adjustable path-length parameters, u≤v. Incorporating quantum concurrence percolation and comparing it with classical percolation, we analyze the transitions of critical thresholds for various values of f and v from analytical, numerical, and simulation perspectives. The results demonstrate that quantum percolation exhibits consistently lower critical thresholds than classical percolation, even under various topologies and dilution levels. Our work implies that quantum multipath entanglement provides an intrinsic compensatory mechanism against structural degradation and that the hierarchical scale-free topology contributes to the failure resistance and robustness of quantum networks with multipath coupling.
Establishing long-distance quantum entanglement, i.e., entanglement transmission, in quantum networks (QN) is a key and timely challenge for developing efficient quantum communication. Traditional comprehension based on classical percolation assumes a necessary condition for successful entanglement transmission between any two infinitely distant nodes: they must be connected by at least a path of perfectly entangled states (singlets). Here, we relax this condition by explicitly showing that one can focus not on optimally converting singlets but on establishing concurrence-a key measure of bipartite entanglement. We thereby introduce a new statistical theory, concurrence percolation theory (ConPT), remotely analogous to classical percolation but fundamentally different, built by generalizing bond percolation in terms of "sponge-crossing" paths instead of clusters. Inspired by resistance network analysis, we determine the path connectivity by series and parallel rules and approximate higher-order rules via star-mesh transforms. Interestingly, we find that the entanglement transmission threshold predicted by ConPT is lower than the known classical-percolation-based results and is readily achievable on any series-parallel networks such as the Bethe lattice. ConPT promotes our understanding of how well quantum communication can be further systematically improved versus classical statistical predictions under the limitation of QN locality-a "quantum advantage" that is more general and efficient than expected. ConPT also shows a percolationlike universal critical behavior derived by finite-size analysis on the Bethe lattice and regular two-dimensional lattices, offering new perspectives for a theory of criticality in entanglement statistics.
Abstract Quantum networks describe communication networks that are based on quantum entanglement. A concurrence percolation theory has been recently developed to determine the required entanglement to enable communication between two distant stations in an arbitrary quantum network. Unfortunately, concurrence percolation has been calculated only for very small networks or large networks without loops. Here, we develop a set of mathematical tools for approximating the concurrence percolation threshold for unprecedented large-scale quantum networks by estimating the path-length distribution, under the assumption that all paths between a given pair of nodes have no overlap. We show that our approximate method agrees closely with analytical results from concurrence percolation theory. The numerical results we present include 2D square lattices of 2002 nodes and complex networks of up to 104 nodes. The entanglement percolation threshold of a quantum network is a crucial parameter for constructing a real-world communication network based on entanglement, and our method offers a significant speed-up for the intensive computations involved.
Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network's indirect connectivity. This realization leads to the emergence of an alternative theory called "concurrence percolation", which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design.
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.
A quantum network is a network of entangled states, which can be used to transmit quantum information. Nonmaximally entangled states are not really effective in establishing quantum communication across vast distances. Creating and maintaining a maximally entangled state over which quantum information can be transferred more effectively is difficult in experiments, as nonmaximally entangled states are typically produced there. Therefore, we usually construct a network of nonmaximally entangled states and use the underlying network structure to establish maximal entanglement between distant nodes. Various protocols leveraging interesting aspects of quantum and statistical physics are designed to achieve long-range maximally entangled states. Here, we consider two-dimensional lattices as quantum networks where edges represent nonmaximally entangled pure states. We introduce a general protocol based on bond percolation -- general concurrence percolation (GCP) -- to create a network of maximally entangled states. The associated network obtained through GCP ought to enable the achievement of long-range quantum communication. Our observations indicate that, similar to other existing protocols, GCP falls within the percolation universality class, as determined through finite-size scaling analysis performed on two-dimensional lattices. We find that the associated percolation threshold value is smaller than that of the existing protocols. We introduce and analytically solve a minimal model that renders insight into the origin of the fundamentally lower percolation threshold achieved by GCP.