Multipartite Entanglement Routing as a Hypergraph Immersion Problem

TL;DR

This paper models multipartite entanglement routing in quantum networks as a hypergraph immersion problem, providing a theoretical framework for understanding topological transformations in quantum network design.

Contribution

It introduces an exact mapping from multipartite entanglement routing to a hypergraph immersion problem, extending graph theory to quantum network topology analysis.

Findings

Hypergraph immersion problem models entanglement routing.

Partial order of network topologies established.

Insights into topological transformations in quantum networks.

Abstract

Multipartite entanglement, linking multiple nodes simultaneously, is a higher-order correlation that offers advantages over pairwise connections in quantum networks (QNs). Creating reliable, large-scale multipartite entanglement requires entanglement routing, a process that combines local, short-distance connections into a long-distance connection, which can be considered as a transformation of network topology. Here, we address the question of whether a QN can be topologically transformed into another via entanglement routing. Our key result is an exact mapping from multipartite entanglement routing to Nash-Williams's graph immersion problem, extended to hypergraphs. This generalized hypergraph immersion problem introduces a partial order between QN topologies, permitting certain topological transformations while precluding others, offering discerning insights into the design and manipulation of higher-order network topologies in QNs.