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.
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.
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.
We study multi-qubit variational quantum states that can be considered as vertex- and edge-weighted graph. These states are constructed as single-layer variational circuits with RX rotations and RZZ entangling gates, corresponding to graphs of arbitrary structure. In general case of quantum graph states of arbitrary structure we derive the geometric measure of entanglement and evaluate quantum correlators. It is shown that these quantities are related to the edge-weight structure around the corresponding vertices in the graph (i.e., edge weights incident to the vertices and vertex weights associated with their closed neighborhoods). In the special case of quantum states representing unweighted graphs, these quantities are related to the degrees of the corresponding vertices in the graph. As an example, we analyze the state associated with the star graph K1,4 using noisy quantum computing on the AerSimulator. The results are in good agreement with theoretical predictions. These findings demonstrate a connection between graph structure and quantum properties, enabling the study of properties of classical graphs via quantum computing.
The hypergraph states are pure multipartite quantum states corresponding to a hypergraph. It is an equal superposition of the states belonging to the computational basis. Given any hypergraph, we can construct a hypergraph state determined by a Boolean function. In contrast, we can find a hypergraph, corresponding to a Boolean function. This investigation develops a number of combinatorial structures concerned with the hypergraph states. For instance, the elements of the computational basis generate a lattice. The chains and antichains in this lattice assist us to find the equation of the Boolean function explicitly as well as to find a hypergraph. In addition, we investigate the entanglement property of the hypergraph states in terms of their combinatorial structures. We demonstrate several classes of hypergraphs, such that every cut of equal length on the corresponding hypergraph states has an equal amount of entanglement.
Abstract The entanglement properties of quantum states associated with directed graphs are investigated. Using a measure derived from the Fubini–Study metric, multipartite entanglement is quantitatively related to the local connectivity of the graph. In Entanglement in Directed Graph States (2025), arXiv:2505.10716, it is demonstrated that the vertex degree distribution fully determines this entanglement measure and remains invariant under vertex relabeling, highlighting its topological character. As a consequence, the measure depends only on the total degree of each vertex, making it independent of the distinction between incoming and outgoing edges. This framework is applied to several specific graph structures, including hierarchical networks, neural network–inspired graphs, full binary tree and linear bridged cycle graphs, demonstrating how their combinatorial properties influence entanglement distribution. These results provide a geometric perspective on quantum correlations in complex systems, offering potential applications in the design and analysis of quantum networks.
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.
Abstract We study the entanglement properties of quantum hypergraph states of n qubits, focusing on multipartite entanglement . We compute multipartite entanglement for hypergraph states with a single hyperedge of maximum cardinality, for hypergraph states endowed with all possible hyperedges of cardinality equal to n − 1 and for those hypergraph states with all possible hyperedges of cardinality greater than or equal to n − 1 . We then find a lower bound to the multipartite entanglement of a generic quantum hypergraph state. We finally apply the multipartite entanglement results to the construction of entanglement witness operators , able to detect genuine multipartite entanglement in the neighbourhood of a given hypergraph state. We first build entanglement witnesses of the projective type, then propose a class of witnesses based on the stabilizer formalism, hence called stabilizer witnesses , able to reduce the experimental effort from an exponential to a linear growth in the number of local measurement settings with the number of qubits.