In this paper we propose two classes of multiparticle entangled states: multigraph and multihypergraph states. Two types of operations are designed to correspond to the edges and hyperedges to prepare the states. Furthermore, multigraph and multihypergraph states contain qudit graph and hypergraph states, respectively. More interestingly, we demonstrate the one-to-one correspondence between the proposed multihypergraph states and the generalized real equally weighted states when the dimension of the states is an odd prime.
In this paper we propose two classes of multiparticle entangled states: multigraph and multihypergraph states. Two types of operations are designed to correspond to the edges and hyperedges to prepare the states. Furthermore, multigraph and multihypergraph states contain qudit graph and hypergraph states, respectively. More interestingly, we demonstrate the one-to-one correspondence between the proposed multihypergraph states and the generalized real equally weighted states when the dimension of the states is an odd prime.
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.
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.
Hypergraph states, a generalization of graph states, constitute a large class of quantum states with intriguing nonlocal properties, and they have promising applications in quantum information science and technology. In this paper, we study some features of an independently proposed generalization of hypergraph states to qudit hypergraph states, i.e., each vertex in the generalized hypergraph (multi-hypergraph) represents a d-level system instead of a two-level one. It is shown that multi-hypergraphs and d-level hypergraph states have a one-to-one correspondence, and the structure of a multi-hypergraph exhibits the entanglement property of the corresponding quantum state. We discuss their relationship with some well-known state classes, e.g., real equally weighted states and stabilizer states. The Bell nonlocality, an important resource in fulfilling many quantum information tasks, is also investigated.
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.
Abstract We study entanglement properties of hypergraph states in arbitrary finite dimension. We compute multipartite entanglement of elementary qudit hypergraph states, namely those endowed with a single maximum-cardinality hyperedge. We show that, analogously to the qubit case, also for arbitrary dimension there exists a lower bound for multipartite entanglement of connected qudit hypergraph states; this is given by the multipartite entanglement of an equal-dimension elementary hypergraph state featuring the same number of qudits as the largest-cardinality hyperedge. We highlight interesting differences between prime and non-prime dimension in the entanglement features.
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.