A resource- and computationally-efficient protocol for multipartite entanglement distribution in Bell-pair networks

Abstract

Multipartite entangled states, particularly Greenberger--Horne--Zeilinger (GHZ) and other graph states, are important resources in multiparty quantum network protocols and measurement-based quantum computing. We consider the problem of generating such states from networks of bipartite entangled (Bell) pairs. We adopt the perspective that, in practice, unlike the traditional information-theoretic setting, local operations and classical communications are not free. Consequently, protocols should not only be efficient with respect to the number of consumed Bell pairs, as typically considered, but also efficient with respect to the number of (local) gates, number of Bell-pair sources, and computational complexity. In this work, we present a protocol for producing GHZ states in arbitrary Bell-pair networks that is efficient with respect to all of these figures of merit. We prove that our protocol: (1) requires $O(N)$ gates in a network with $N$ nodes, independent of the topology of the network; (2) has time complexity $O(N^2)$, thereby avoiding finding a Steiner tree or solving any other computationally hard problem; and (3) maintains nearly the optimal number of consumed Bell pairs. We prove that the minimal Bell-pair source cost is equivalent to the graph-theoretic dominating set problem, and via numerical simulations on real-world network models, we demonstrate that our protocol is nearly optimal with respect to Bell-pair source cost. Numerically, our protocol also outperforms existing protocols based on (approximate) Steiner trees with respect to both number of gates and Bell-pair sources. Finally, we provide a detailed analysis of the impact of noisy Bell pairs and gates on the fidelity of the distributed GHZ states.