# Simulation of Fidelity in Entanglement-Based Networks with Repeater Chains
## 7 Results  
In this section, the results of the experiments will be discussed. As stated before, fidelity will be the key performance metric to assess each of the experiments and topologies. The discussed error models aim at obtaining the most precise fidelity possible given the physical setting, as this metric is not measurable in real setups but it is the most relevant property of the entangled states for communication. Firstly, we will compare two types of quantum architectures, PS and SP [18], as stated in Section 6.1. The former uses purification before entanglement swapping (PS), whereas the opposite strategy is used in SP. With the models proposed, the fidelity values between the architectures were similar for distances up to 70 km, as can be seen in Figure 6. At that point, the PS protocol becomes better at the cost of less probability of achieving a successful purification (more time consumed). Because of this fidelity increase, we will proceed with performing purification of the qubits and then the swapping operations (PS) with correction gates applied at the end node qubit. We implemented the simulation scenarios in Section 6.1 (multi-hop with a fixed number of memory positions) and Section 6.2 (two-hop path with a variable number of memory positions) to determine the effectiveness of the system and establish the minimal resources needed to achieve practical quantum communication. Both scenarios are observed in the presence of a depolarization error in quantum channels, which allows the study of the influence of the number of repeaters and memory positions; other error models will produce similar results but with shorter distances. For instance, in Figure 7a, we evaluate the network architecture discussed in Section 6.1 for a network of total distance \(L=1000\) km, four memory positions for each node, and a source delay of 1 ns. All sources of error have been suppressed except for channel depolarization to give a better understanding of the implemented depolarization model. In this case, without purification, we see an inflection point at around 10 nodes, which corresponds to an internode distance of 125 km. This should be the upper limit for quantum channels in terms of fidelity, as longer channels would rapidly decrease the fidelity and be out of bounds for proper purification. Additionally, close to the same 10-node mark, there is a fidelity value of \(F=0.95\), which the authors consider to be the minimum usable fidelity for applications, specifically without purification. Figure 5: Architecture discussed in Section 6.2. A two-hop network is established, and the program arranges and identifies the fidelity of the \(m=18\) entangled pairs for each of the connections. The route with the highest fidelity is selected to drive the communication. On the other hand, for the scenario in Section 6.2, we can evaluate how fidelity behaves when varying the amount of memory positions. In Figure 7b, we can see for a network of 1000 km how the fidelity increases with the number of memory positions. In a real scenario, attenuation of the signals would be the primary error source and this behavior could not be appreciated, but we wanted to focus on the depolarization model for these experiments. As we can see in Figure 8, we expect an improvement in the high percentile of the fidelity regions. This concludes that the theoretical advantage of the DEJMPS is also present in the simulations; although it consists of a rough 1%, the protocol has only been applied once. We now can compare these results by increasing the number of memory positions within a node (Figure 7b), which increases the chances of establishing high-fidelity links  
Figure 6: (Color online) Fidelity average for the two different protocols discussed, PS and SP. These simulations have been performed in a network of 3 nodes. Error bars correspond to 1 standard deviation. Figure 7: Comparison of the impact on fidelity by varying the number of intermediate QR (**a**) and by varying the number of memory positions in a QR (**b**).