Percolation theory is widely used to analyze the robustness of quantum networks. However, most studies focus on site-bond percolation in quantum network graphs based on bipartite entanglement and entanglement swapping, overlooking the higher-order interactions necessary to model dynamic quantum networks that include propagating multipartite entangled states. Integrating these elements into a hypergraph percolation framework represents a current gap in the scholarly literature. In hypergraphs, while hyperedge percolation generally results in larger giant components than node percolation, we hypothesize that multipartite (hyperedge) percolation can achieve greater connectivity than bipartite (edge) percolation under suitable conditions. Our results show that when the probability of multipartite entanglement remaining intact is low, distributing only bipartite entanglement provides better connectivity due to lower memory overhead. Conversely, when hyperedge generation is reliable and measurements at repeaters are probabilistic, multipartite entanglement enables networks to achieve full connectivity even at low success probabilities, unlike bipartite networks, which require near-perfect success to percolate.
Percolation theory is widely used to analyze the robustness of quantum networks. However, most studies focus on site-bond percolation in quantum network graphs based on bipartite entanglement and entanglement swapping, overlooking the higher-order interactions necessary to model dynamic quantum networks that include propagating multipartite entangled states. Integrating these elements into a hypergraph percolation framework represents a current gap in the scholarly literature. In hypergraphs, while hyperedge percolation generally results in larger giant components than node percolation, we hypothesize that multipartite (hyperedge) percolation can achieve greater connectivity than bipartite (edge) percolation under suitable conditions. Our results show that when the probability of multipartite entanglement remaining intact is low, distributing only bipartite entanglement provides better connectivity due to lower memory overhead. Conversely, when hyperedge generation is reliable and measurements at repeaters are probabilistic, multipartite entanglement enables networks to achieve full connectivity even at low success probabilities, unlike bipartite networks, which require near-perfect success to percolate.
Building large-scale quantum computers, essential to demonstrating quantum advantage, is a key challenge. Quantum networks can help address this challenge by enabling the construction of large, robust, and more capable quantum computing platforms by connecting smaller quantum computers. Moreover, unlike classical systems, quantum networks can enable fully secured long-distance communication. Thus, quantum networks lie at the heart of the success of future quantum information technologies. In quantum networks, multipartite entangled states distributed over the network help implement and support many quantum network applications for communications, sensing, and computing. Our work focuses on developing optimal techniques to generate and distribute multipartite entanglement states efficiently. Prior works on generating general multipartite entanglement states have focused on the objective of minimizing the number of maximally entangled pairs while ignoring the heterogeneity of the network nodes and links as well as the stochastic nature of underlying processes. In this work, we develop a hypergraph-based linear programming framework that delivers optimal (under certain assumptions) generation schemes for general multipartite entanglement represented by graph states, under the network resources, decoherence, and fidelity constraints, while considering the stochasticity of the underlying processes. We illustrate our technique by developing generation schemes for the special cases of path and tree graph states and discuss optimized generation schemes for more general classes of graph states. Using extensive simulations over a quantum network simulator, we demonstrate the effectiveness of our developed techniques and show that they outperform prior known schemes by up to orders of magnitude.
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.
Entanglement percolation aims at generating maximal entanglement between any two nodes of a quantum network by utilizing strategies based solely on local operations and classical communication between the nodes. As it happens in classical percolation theory, the topology of the network is crucial, but also the entanglement shared between the nodes of the network. In a network of identically partially entangled states, the network topology determines the minimum entanglement needed for percolation. In this work, we generalize the protocol to scenarios where the initial entanglement shared between any two nodes of the network is not the same but has some randomness. In such cases, we find that for classical entanglement percolation, only the average initial entanglement is relevant. In contrast, the quantum entanglement percolation protocol (within the q-swap framework) degrades under these more realistic conditions as the width of the distribution increases, suggesting that Random CEP may become the optimal LOCC strategy in sufficiently heterogeneous quantum networks.
Quantum networks have good prospects for applications in the future. Compared with classical networks, small-world quantum networks have some interesting properties. The topology of the network can be changed through entanglement exchange operations, and different network topologies will result in different percolation thresholds when performing entanglement percolation. A lower percolation threshold means that quantum networks require fewer minimum resources for communication. Since a shared singlet between two nodes can still be a limitation, concurrency percolation theory (ConPT) can be used to relax the condition. In this paper, we investigate how entanglement distribution is performed in small-world quantum networks to ensure that nodes in the network can communicate with each other by establishing communication links through entanglement swapping. Any node can perform entanglement swapping on only part of the connected edges, which can reduce the influence of each node in the network during entanglement swapping. In addition, the ConPT method is used to reduce the percolation threshold even further, thus obtaining a better network structure and reducing the resources required.
Quantum hypergraph states extend the well-studied class of graph states by taking into account multi-qubit interactions through hyperedges. They provide a powerful framework to represent a family of quantum states with genuine multipartite entanglement. In this review, we provide a compact overview of the formal structure, entanglement characteristics, and operational relevance of hypergraph states in quantum information theory. We begin by introducing their mathematical foundations and generalizations of the stabilizer formalism. A central focus is placed on their entanglement properties, including the classification under local unitary (LU) and stochastic local operations with classical communication (SLOCC), the quantification of multipartite entanglement, and detection techniques via entanglement witnesses. We also explore other nonclassical features of hypergraph states, such as contextuality and genuine multipartite nonlocality, derived from stabilizer-based Bell-type inequalities. Additional attention is given to the role of hypergraph states in error correction, and as a computational resource in measurement-based quantum computation (MBQC), and to their non-stabilizer character - quantified via resource-theoretic measures of quantum magic. Finally we review their generalization to higher dimensions, i.e. to qudits and continuous variables.
Hypergraphs generalize ordinary graphs by allowing hyperedges to join any number of vertices rather than just two. Superhypergraphs extend this idea further by iterating the powerset construction, creating hierarchical, nested hyperedge structures. In parallel, quantum graphs model networks of quantum systems, with wavefunctions propagating along edges under specified boundary conditions. Quantum hypergraph states translate hypergraph connectivity into multi-qubit entanglement via generalized controlled phase gates acting on all vertices of each hyperedge. In this paper, we introduce quantum n-superhypergraph states, which marry the recursive structure of superhypergraphs with the formalism of quantum hypergraph states. We give their precise definition, explore key structural properties, and outline potential directions for their application.