A Theoretical Investigation of Quantum n-SuperHypergraph States

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.