Quantum Hypergraph States: A Review

Abstract

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.

Figures (12)

• Figure 1: Example for a graph state: Representation of a graph state of $5$ qubits and the corresponding quantum circuit. The state is initialized as $\ket{+}^{\otimes 5}$, and a $\mathrm{CZ}$ gate is applied for every edge in the graph $G_5$. The $\mathrm{CZ}$ gates are represented by a line, with dots indicating qubits on which they are acting.
• Figure 2: Example for a hypergraph state: Representation of a hypergraph state of $5$ qubits with edges of cardinality $2$ and $3$, and the corresponding quantum circuit. The state is initialized as $\ket{+}^{\otimes 5}$, and for every $2-$ or $3$-edge in the graph $H_5$ a $\mathrm{CZ}$ or $\mathrm{C}^{3}\mathrm{Z}$ gate is applied, respectively. The gates are represented by a line, with dots indicating qubits on which they are acting.
• Figure 3: Various examples of (hyper)graph states:\ref{['fig:hg_example_gc3']} and \ref{['fig:hg_example_gstar']} examples of standard graph states on $3$ vertices, while \ref{['fig:hg_example_gcluster']} is a cluster state on $8$ vertices. \ref{['fig:hg_example_hcl3']} is the simplest hypergraph state, having only one global $3$-edge containing all the nodes. Together with \ref{['fig:hg_example_triskell']} and \ref{['fig:hg_example_hcl5']} it is part of the family of Clover hypergraphs with $3,4$ and $5$ vertices respectively, which are $k$-uniform subgraphs sharing a vertex in all the edges. \ref{['fig:hg_example_hcl4']} is an example of a complete (or symmetric) $3$-uniform hypergraph. \ref{['fig:hg_example_hcl34']} instead is also symmetric but not $3$-uniform, since it contains also a global $4$-edge.
• Figure 4: Hypergraph, REW, and LME States:
• Figure 5: Graphical Pauli rules: Examples for the graphical rules for the action of local Pauli operators on a hypergraph state. In the top row, the action of $X_1$ on the first qubit creates two new $2$-edges, and one $1$-edge on qubit 4 since these edges are not present in the original hypergraph. In the bottom row, acting with $X_2$ on qubit $2$ deletes the edge $(1,4)$ instead. Conversely the action of $Z$ can only remove or add a single $1$-edge on the index where it is applied.

Theorems & Definitions (8)

• Definition 1: Graph state
• Definition 2: Hypergraph state
• Definition 3: Stabilizer states
• Theorem 1: Knill--Laflamme (KL) condition
• Definition 4: Multihypergraph
• Definition 5: Multihypergraph states
• Definition 6: CV graph state
• Definition 7: CV hypergraph state