ABSTRACT We study the entanglement properties of randomized mixed hypergraph states, extending the concept of randomized mixed graph states to encompass hypergraph‐based quantum states. In our model, imperfect generalized multi‐qubit gates are applied probabilistically, simulating experimentally realistic noisy gate operations where gate fidelity decreases with increasing hyperedge order. We analyze bipartite and genuine multipartite entanglement of these mixed multi‐qubit states. Numerical results for various hypergraph configurations with up to four qubits reveal rich, sometimes nonmonotonic entanglement behavior stemming from the interplay between hyperedge structure and gate imperfections. We derive analytical expressions for entanglement witnesses based on randomization overlap for new hypergraph families. Our findings contribute to understanding entanglement resilience under gate imperfections, providing insight into the experimental implementation of hypergraph states in noisy quantum devices.
ABSTRACT We study the entanglement properties of randomized mixed hypergraph states, extending the concept of randomized mixed graph states to encompass hypergraph‐based quantum states. In our model, imperfect generalized multi‐qubit gates are applied probabilistically, simulating experimentally realistic noisy gate operations where gate fidelity decreases with increasing hyperedge order. We analyze bipartite and genuine multipartite entanglement of these mixed multi‐qubit states. Numerical results for various hypergraph configurations with up to four qubits reveal rich, sometimes nonmonotonic entanglement behavior stemming from the interplay between hyperedge structure and gate imperfections. We derive analytical expressions for entanglement witnesses based on randomization overlap for new hypergraph families. Our findings contribute to understanding entanglement resilience under gate imperfections, providing insight into the experimental implementation of hypergraph states in noisy quantum devices.
We introduce and analyze the entanglement properties of randomized hypergraph (RH) states, as an extended notion of the randomization procedure in the quantum logic gates for the usual graph states, recently proposed in the literature. The probabilities of applying imperfect generalized controlled-Z gates simulate the noisy operations over the qubits. We obtain entanglement measures as negativity, concurrence, and genuine multiparticle negativity, and show that entanglement exhibits a nonmonotonic behavior in terms of the randomness parameters, which is a consequence of the nonuniformity of the associated hypergraphs, reinforcing the claim that the entanglement of randomized graph states is monotonic since they are related to 2-uniform hypergraphs. Moreover, we observed the phenomena of entanglement sudden death and entanglement sudden birth in RH states. This work reveals a connection between the nonuniformity of hypergraphs and loss of entanglement.
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.
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.
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.
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.