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.
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.
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.
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 and stochastic local operations with classical communication, 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, 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.
We propose a method for constructing multi-qubit entangled quantum states that represent weighted tripartite graphs, and develop approaches for investigating their structural properties using quantum computing. In the general case of multi-qubit states corresponding to arbitrary tripartite graph structures, we derive an expression for the entanglement distance. We establish a connection between entanglement and properties of the corresponding tripartite graphs. Namely, we show that the entanglement of a qubit with the rest of the system in a quantum tripartite graph state depends on the weights of the arcs in the neighborhood of the corresponding vertex, as well as on its degree with respect to the vertex sets of the tripartite graph. As an illustrative example, we consider a tripartite graph forming a triangle and evaluate the entanglement distance with quantum computing. We also compute quantum correlators for the general case of tripartite quantum graph states and relate these quantities to structural features of the underlying graphs, including the number of non-overlapping neighbors, the number of common neighbors of the corresponding vertices, and the number of 4-cycles. Obtained relationships between the quantum properties of multi-qubit states and the structural features of tripartite graphs opens up the possibility of investigating classical systems, such as tripartite graphs, using quantum computing. It is worth emphasizing that tripartite graphs have applications in practical problems, including resource allocation, scheduling, and database and hypergraph modeling.
The encoding of 3-uniform edge-ordered weighted hypergraphs into multiqubit quantum states is proposed. We derive an analytical expression for the geometric measure of entanglement of hypergraph quantum states and establish its relation to hypergraph properties. The results establish a direct connection between the local structure of hypergraphs, hyperedge weights, and the entanglement of the corresponding quantum states. For equally weighted hypergraphs, it is shown that the entanglement depends explicitly on the vertex degree. The dependence of entanglement on hypergraph parameters is studied analytically and using quantum computing. Hypergraph states associated with chain, star, regular lattice, and binary-tree hypergraphs are examined, and their entanglement is quantified using quantum computing. These results open up the possibility of studying hypergraph properties with quantum programming.
Magic, or nonstabilizerness, characterizes the deviation of a quantum state from the set of stabilizer states, playing a fundamental role in quantum state complexity and universal fault-tolerant quantum computing. However, analytical and numerical characterizations of magic are very challenging, especially for multi-qubit systems, even with a moderate qubit number. Here, we systemically and analytically investigate the magic resource of archetypal multipartite quantum states—quantum hypergraph states, which can be generated by multi-qubit controlled-phase gates encoded by hypergraphs. We first derive the magic formula in terms of the stabilizer Rényi- α entropies for general quantum hypergraph states. If the average degree of the corresponding hypergraph is constant, we show that magic cannot reach the maximal value, i.e., the number of qubits n . Then, we investigate the statistical behaviors of random hypergraph states' magic and prove a concentration result, indicating that random hypergraph states typically reach the maximum magic. This also suggests an efficient way to generate maximal magic states with random diagonal circuits. Finally, we study hypergraph states with permutation symmetry, such as 3 -complete hypergraph states, where any three vertices are connected by a hyperedge. Counterintuitively, such states can only possess constant or even exponentially small magic for α≥2 . Our study advances the understanding of multipartite quantum magic and could lead to applications in quantum computing and quantum many-body physics.
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.