Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1612.06418 · Physical Review A

Qudit hypergraph states

Frank E. S. Steinhoff, Christina Ritz, Nikolai Miklin, Otfried Gühne

Published May 22, 2017• 19 citations
arXivHTMLPDF

Abstract

We generalize the class of hypergraph states to multipartite systems of qudits, by means of constructions based on the $d$-dimensional Pauli group and its normalizer. For simple hypergraphs, the different equivalence classes under local operations are shown to be governed by a greatest-common-divisor hierarchy. Moreover, the special cases of three qutrits and three ququarts is analyzed in detail.

DOI
10.1103/physreva.95.052340

Related papers

Multipartite entanglement and hypergraph states of three qubits

Ri Qu, Zong-shang Li, Juan Wang +1 more

Several entanglement measures are used to define equivalence classes in the set of hypergraph states of three qubits. Our classifications reveal that (i) under local unitary transformations, hypergraph states of three qubits are split into six classes and only one of them is not equivalent to any graph state; (ii) under stochastic local operations with classical communication, for the single-copy case hypergraph states of three qubits, partitioned into five classes which cannot be converted into a $W$ state, are equivalent to graph states; and (iii) when bipartite entanglement in three qubits is considered, hypergraph states of three qubits are split into five classes and only one of them has the same entangled graph as the $W$ state.

arXiv:1301.3576•Physical Review A•Mar 25, 2013•13 citations

Encoding hypergraphs into quantum states

Ri Qu, Juan Wang, Zong-shang Li +1 more

Ionicioiu and Spiller [Phys. Rev. A 85, 062313 (2012)] have recently presented an axiomatic framework for mapping graphs to quantum states of a suitable physical system. Based on their study, we first extend the axiomatic framework to hypergraphs by means of modifying its axioms and consistency conditions. Then we use the axiomatic approach to encode hypergraphs into a different family of quantum states, called the hypergraph states. Moreover, we also try to do the following: (i) show that real equally weighted states, which occur in Grover and Deutsch-Jozsa algorithms, are equivalent to hypergraph states; (ii) describe the relations among hypergraph states, graph states, and stabilizer states; (iii) provide some transformation rules, stated in purely hypergraph theoretical terms, which completely characterize the evolution of hypergraph states under some local operations, including operators in the Pauli group and some special local Pauli measurements; and (iv) investigate some properties of multipartite entanglement of hypergraph states by hypergraph theory.

arXiv:1211.3911•Physical Review A•Feb 13, 2013•63 citations

Multipartite entanglement in qudit hypergraph states

Daniele Malpetti, Alfredo Bellisario, Chiara Macchiavello

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.

arXiv:2205.08465•Journal of Physics A Mathematical and Theoretical•Sep 13, 2022•2 citations

Quantum Hypergraph States: A Review

Davide Poderini, Dagmar Bruß, Chiara Macchiavello

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.

quant-pharXiv:2603.10917•Mar 12, 2026•0 citations

Calibrated hypergraph states. II. Calibrated hypergraph state construction and applications

Roberto Zucchini

Hypergraph states are a special kind of multipartite states encoded by hypergraphs relevant in quantum error correction, measurement–based quantum computation, quantum non locality and entanglement. In a series of two papers, we introduce and investigate calibrated hypergraph states, an extension of weighted hypergraph states codified by hypergraphs equipped with calibrations, a broad generalization of weightings. The guiding principle informing our approach is that a constructive theory of hypergraph states must be based on a categorical framework for both hypergraphs and multi qudit states constraining hypergraph states enough to render the determination of their general structure possible. In this second paper, we build upon the graded Ω monadic framework worked out in the companion paper, focusing on qudits over a generic Galois ring. We explicitly construct a calibrated hypergraph state map as a special morphism of the calibrated hypergraph and multi qudit state Ω monads. We further prove that the calibrated hypergraph states so yielded are locally maximally entangleable stabilizer states, elucidate their relationship to weighted hypergraph states, show that they reduce to the weighted ones in the familiar qubit case and prove through examples that this is no longer the case for higher qudits.

arXiv:2501.18968•Journal of Mathematical Physics•Jul 1, 2026•0 citations

Graphical description of unitary transformations on hypergraph states

Mariami Gachechiladze, Nikoloz Tsimakuridze, Otfried Gühne

Abstract Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application, we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for three-uniform hypergraph states.

arXiv:1612.01447•Journal of Physics A Mathematical and Theoretical•Mar 17, 2017•13 citations
We generalize the class of hypergraph states to multipartite systems of qudits, by means of constructions based on the ddd-dimensional Pauli group and its normalizer. For simple hypergraphs, the different equivalence classes under local operations are shown to be governed by a greatest-common-divisor hierarchy. Moreover, the special cases of three qutrits and three ququarts is analyzed in detail.
Several entanglement measures are used to define equivalence classes in the set of hypergraph states of three qubits. Our classifications reveal that (i) under local unitary transformations, hypergraph states of three qubits are split into six classes and only one of them is not equivalent to any graph state; (ii) under stochastic local operations with classical communication, for the single-copy case hypergraph states of three qubits, partitioned into five classes which cannot be converted into a WWW state, are equivalent to graph states; and (iii) when bipartite entanglement in three qubits is considered, hypergraph states of three qubits are split into five classes and only one of them has the same entangled graph as the WWW state.
Ionicioiu and Spiller [Phys. Rev. A 85, 062313 (2012)] have recently presented an axiomatic framework for mapping graphs to quantum states of a suitable physical system. Based on their study, we first extend the axiomatic framework to hypergraphs by means of modifying its axioms and consistency conditions. Then we use the axiomatic approach to encode hypergraphs into a different family of quantum states, called the hypergraph states. Moreover, we also try to do the following: (i) show that real equally weighted states, which occur in Grover and Deutsch-Jozsa algorithms, are equivalent to hypergraph states; (ii) describe the relations among hypergraph states, graph states, and stabilizer states; (iii) provide some transformation rules, stated in purely hypergraph theoretical terms, which completely characterize the evolution of hypergraph states under some local operations, including operators in the Pauli group and some special local Pauli measurements; and (iv) investigate some properties of multipartite entanglement of hypergraph states by hypergraph theory.
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.
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 are a special kind of multipartite states encoded by hypergraphs relevant in quantum error correction, measurement–based quantum computation, quantum non locality and entanglement. In a series of two papers, we introduce and investigate calibrated hypergraph states, an extension of weighted hypergraph states codified by hypergraphs equipped with calibrations, a broad generalization of weightings. The guiding principle informing our approach is that a constructive theory of hypergraph states must be based on a categorical framework for both hypergraphs and multi qudit states constraining hypergraph states enough to render the determination of their general structure possible. In this second paper, we build upon the graded Ω monadic framework worked out in the companion paper, focusing on qudits over a generic Galois ring. We explicitly construct a calibrated hypergraph state map as a special morphism of the calibrated hypergraph and multi qudit state Ω monads. We further prove that the calibrated hypergraph states so yielded are locally maximally entangleable stabilizer states, elucidate their relationship to weighted hypergraph states, show that they reduce to the weighted ones in the familiar qubit case and prove through examples that this is no longer the case for higher qudits.
Abstract Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application, we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for three-uniform hypergraph states.