# A Performance-Consumption Balanced Scheme of Multi-Hop Quantum Networks for Teleportation
## 4 Design of Purification Algorithm and Simulation Results
### Simulation Results  
The simulation result of the 5-node network in Figure 5 is shown in Figure 10. We should point out that the simulation is influenced by both channel noise and fidelity loss in entanglement swapping. Entanglement swapping does not transfer qubit in the channel, but it surly brings a huge fidelity loss. Thus, we set the fidelity loss at a fixed value of 12.77% in entanglement swapping. Because of the fixed fidelity loss, the system fidelity cannot reach 1 even without noise in the channel. In Figure 10, the blue line represents the 5-node network based on Bell states without purification. After purification, the orange line represents the Bell states scheme only using Bell states, the purple line represents the GHZ states scheme only using GHZ states, and the yellow line represents the fusion scheme we have proposed. The purification algorithm can improve the fidelity of the final entangled states. When \(p\) is small, the fusion scheme and GHZ states scheme are far better than the Bell states scheme under bit-flip noise. However, under depolarizing noise, the advantage of the GHZ states scheme is not obvious, which is even worse than the Bell states scheme most of the time. We can treat depolarizing noise as a mixed noise of bit-flip and phase-flip. In purification of Bell states, there is no effective way to solve the noise with bit-flip and phase-flip together. In addition, in purification of GHZ states, there is no effective way to solve the presence of phase-flip noise. Under bit-flip noise, there is no need to consider the existence of phase-flip noise. Thus, the purification performance under depolarizing noise is most of the time worse than that under bit-flip noise. However, the fusion scheme we have proposed is surely better than the Bell states scheme. The performance of the fusion scheme is close to that of the GHZ states scheme under bit-flip noise. Under depolarizing noise, the fusion scheme is the best. In the purification scheme we have introduced, each successful purification needs to measure one entangled group. Entanglement swapping also needs to measure lots of entangled states. Purification does not always succeed, which will waste more entanglement. To measure the resource consumption in different schemes, we calculate the minimum basic states required in Algorithm 1. We generate one Bell pair with two basic states based on a H gate and CNOT gate. Although we obtain a GHZ group in the fusion scheme by using two Bell pairs, we use 3 basic states to get the entangled group in the GHZ states scheme. Figure 10: Fidelity as a function of bit-flip and depolarizing noise with purification. Purification is applied after every entanglement swapping with time of \(m_{1}=1\). After the entanglement is established in the source and distribution node, extra purification is applied with time of \(m_{2}=1\). Considering that measuring only one qubit in the GHZ group leads to the disappearance of the whole entanglement, entanglement swapping in the GHZ states scheme needs to measure two qubits in one group at a time to keep the system entangled. The number of states needed in the network with different node numbers is shown in Table 3. The fusion scheme improves the network performance in comparison to the Bell states scheme and consumes less qubits than the GHZ states scheme. Simulation is performed for networks spanning different numbers of nodes and is shown in Figure 11. With increasing number of nodes, the fidelity decreases. The performance of the fusion network is significantly better than that of the Bell states scheme. In the fusion scheme, the calculation complexity is \(O(1)\). After entanglement swapping and purification, high-fidelity entanglement has been established between source and destination node. One can directly use the entangled group to finish teleportation at last. The price is sacrificing a lot of entanglement resources in purification and entanglement swapping. To verify the scheme in the true environment, we finish the simulation on the NISQ platform Cirq.