Routing Entanglement in Complex Quantum Networks Using GHZ States

Abstract

Distributing entanglement to distant parties in a network is a central task in quantum information processing and quantum networking. The sensitivity of entangled states to loss necessitates the use of entanglement routing strategies. Recently, a routing strategy using Greenberger-Horne-Zeilinger (GHZ) measurements instead of Bell state measurements (BSM) has been proposed. We further this direction of research by explicitly considering the varying measurement success probabilities of GHZ measurements. Moreover, we extend the analysis beyond square grid networks to complex network models such as Waxman networks and scale-free networks, as well as SURFnet, a real-world network topology in the Netherlands. Taking into account the varying success probabilities, naive application of GHZ routing achieves rates much lower than the conventional BSM routing. Instead, we propose a hybrid GHZ-BSM routing strategy. The hybrid GHZ-BSM routing strategy outperforms BSM routing in square grid networks. In other networks, however, more sophisticated adaptations using global information are required.

Figures (8)

• Figure 1: (a) Example of a physical Waxman network with 30 nodes on a 100 km $\times$ 100 km square. We chose $\alpha=1.598$ so that $\alpha L=226$ km, and $\beta=1$. (b) Example of a scale-free network with 30 nodes on a 100 km $\times$ 100 km square. We chose $m=5$, $\mu=\nu=1$, and the initial network is a 6-node complete graph. (c) SURFnet topology
• Figure 2: (a) Example of a virtual Waxman network. (b) Example of a virtual scale-free network. The gray lines represent physical links, while the blue lines represent the successfully established entanglement links between two adjacent nodes. For both, we have assumed that the fiber-optic loss is given by $\gamma=0.2$ km$^{-1}$.
• Figure 3: Illustration of the output states after a Bell state measurement
• Figure 4: (a) Illustration of our proposed GHZ state protocol on a square grid. The blue box represents one of the two end users, the gray boxes represent helper nodes, the green shaded region represents the prepared states, and the orange boxes represent a Bell state measurement on the two received photons. (b) Illustration of the original GHZ measurement protocol patil2022entanglementkaur2023distribution. The blue box represents one of the two users, and the gray boxes represent helpers that perform $l$-GHZ measurements, where $l$ is the number of qubits that is received by the user.
• Figure 5: Performance evaluation for square grid networks. (a) Average rate $\langle R\rangle$ for an $N\times N$ grid for $N$ from 8 to 17. (b--d) Rate vs. distance (in km) for $N=15$ and $q=0.7,0.8,0.9$, respectively.