Entanglement routing in hexagonal lattice hypergraph quantum networks via 2-localization and hypergraph immersion

Abstract In this paper, we investigate entanglement routing in hexagonal lattice hypergraph quantum networks (HLHQNs) under 2-localization and hypergraph immersion. Firstly, we model each hexagonal face as a six-partite GHZ hyperedge, and show that each hexagonal GHZ hyperedge admits deterministic 2-localization, namely, it can be converted via local operations and classical communication into a Bell state between any designated pair of vertices. This property is further generalized to arbitrary k -partite GHZ states. Based on this GHZ-specific localization property, we develop two routing paradigms: localization-then-swapping (L–S) and coalescence-then-dewetting (C–D). Then, under hypergraph immersion, we formulate bipartite routing feasibility as an edge-disjoint path immersion problem on the effective localized graph, and derive illustrative scaling analyzes for both paradigms. Finally, numerical simulations on finite HLHQNs reveal distinct operational mechanisms, resource overheads and applicable scenarios. The L–S paradigm excels at point-to-point bipartite communication and concurrent routing, while the C–D paradigm is more efficient for direct multipartite GHZ-state construction over compact target sets. Our results establish a routing framework specifically for HLHQNs with GHZ-type hyperedge resources and may provide theoretical support for the design of scalable high-order entanglement quantum networks.