# A Performance-Consumption Balanced Scheme of Multi-Hop Quantum Networks for Teleportation
## 1 Introduction  
Teleportation is always used for transferring quantum information with the aid of maximally entangled states and classical channels. In 1993, [PERSON] proposed the first teleportation protocol based on the Einstein-Podolsky-Rosen (EPR) state [1]. In 1997, [PERSON] proved it in the practical experiment [2]. Since then, many other teleportation protocols were proposed based on different entangled states, such as GHZ states, W states, and cluster states [3; 4; 5]. Teleportation can be easily realized between two adjacent nodes. However, in a multi-hop quantum network, most nodes are not directly connected. Realizing teleportation in a multi-hop quantum network is valuable. In 2005, [PERSON] proposed the Bridging protocol of the network in this field [6], and then, the ideas for teleportation network were given in different entangled states [7; 8]. However, they are all designed based on the teleportation between directly connected nodes. [PERSON] gave a classical solution to it [9]. Quantum states in all intermediate nodes are measured in the end to realize teleportation. Increasing the number of nodes between the source and the destination will bring difficulties with exponential growth to the calculation. Besides, the influence of noise is also amplified. As an important technology in quantum relay, entanglement swapping has been realized in the recent years [10; 11; 12; 13]. The original entanglement swapping protocol took advantage of the spontaneous parametric down-conversion (SPDC) source [14]. Today, scientists prefer to use hybrid protocols and photon interference to entangle spins or atoms [15; 16]. Scientists have proved that entanglement swapping can establish entanglement between states in different entangled pairs. The price is the loss of fidelity [17; 18]. This technology makes it possible for arbitrary nodes in the network to share entangledstates. When it comes to the noise in the environment, scientists think that purification can solve this problem [19; 20; 21]. Because of a developed purification [22; 23; 24; 25], high fidelity entanglement can be established in the multi-hop quantum network. Based on the entanglement swapping and purification, we design a new scheme of a multi-hop quantum network. Taking advantage of entanglement swapping and purification, nodes in the network can share entangled states with high fidelity. To ensure the high robustness of the system, we design a purification algorithm. To ensure the practicability of the system, we use bit-flip, phase-flip, and depolarizing noise models in the simulation. Although amplitude damping noise seems more practical in the simulation, we find that it will lead to the disappearance of entanglement. Considering purification always works on entangled states, thus, we did not use amplitude damping noise in this paper. Besides, we consider the decline in fidelity of entanglement swapping based on the latest physical experiment. The number of entering nodes is decreased in the final communication to protect the privacy. Thus, the computational complexity is reduced. In the end, we simulate the network on Cirq with bit-flip and depolarizing noise in the true environment. Contributions of this paper can be stated as follows:  
* This paper proposes a new scheme of a multi-hop quantum network based on bipartite communication with fusion states. It improves the noise immunity of the network while avoiding the huge consumption. * A purification algorithm is proposed to ensure the robustness of the system, providing a solution for future quantum communication. * The fidelity for multi-particle entanglement purification is calculated. The paper is outlined as follows. In Section 2, we review entangled states, teleportation, entanglement swapping, quantum noise, and fidelity. In Section 3, we design the new multi-hop quantum network architecture based on bipartite communication. Besides, we give the purification scheme for the Bell pair and GHZ states. In Section 4, we give the design for a purification algorithm and simulation results.