# Simulation of Fidelity in Entanglement-Based Networks with Repeater Chains
## 1 Introduction  
Long-distance quantum communications and distributed quantum computing tasks necessitate efficient entanglement distribution among the parties to be feasible on a practical scale in order to process quantum information in better and more capable ways than those achievable with classical communications [1; 2; 3]. Also, high-rate entanglement distribution is key for realizing information-theoretical secure communications [4]. However, long-distance distribution of entangled states requires the use of quantum repeaters (QR) or quantum switches (QS) as intermediate devices to sequentially propagate the link-level entanglement [5; 6] so that end-to-end entangled (EEE) states between two arbitrary nodes are created. Functionally, a QR and a QS are similar, but QRs are simpler since they are basically two-port devices that extend a linear chain of links, while QSs could, in principle, allow for more general internal quantum connections from an input to an output port [7]. As in classical networking, QRs would enable the construction of general physical interconnection topologies for a quantum network. Controlling the delay, throughput, and quality (in fidelity, a quantitative measure of entanglement) of a quantum network based on repeaters or switches is thus a fundamental challenge for practical applications [8; 9]. Different analytical models are being developed for elucidating the performance of quantum networks under a variety of assumptions. Though the insights and design principles gained with analytical models are invaluable to understanding the complex relationships between physical parameters and network performance, the known analytical tools can only solve simple scenarios, are restricted to configurations with few nodes, or consider quantum repeaters or specific entanglement protocols, e.g., refs. [10; 11; 12]. Furthermore, these works rely on idealized assumptions about the diverse impairments that quantum devices or quantum channels may exhibit, e.g., loss, state/memory decoherence, noisy entanglement swapping, or depolarization errors. Given the fragility of these basic quantum information processing tasks, and the breadth of physical parameters that underlie the behavior of quantum devices and channels, tailored quantum-network simulators are often used in numerical studies [13; 14; 15; 16] for two reasons: (i) to obtain a precise evaluation of non-trivial network configurations when the number of physical variables required to accurately model each QR and the number of intermediate nodes is too large; (ii) to test and compare, in a quick experiment, different combinations of architectures and protocols suitable for a given computational goal. There exist many algorithms, for instance, for distributing entangled states [17] and using purification vs. error correction at the nodes [18], entanglement swapping protocols [19], routing protocols [20], etc. For the numerical characterization of a quantum network, the throughput and fidelity of the EEE link are arguably the key performance indicators. The throughput is defined as the rate of successful entangled states created between two endpoints and depends on the physical network constraints, the quantum resource constraints (the number of memory qubits at the QR), and the embedded protocols. Fidelity is a measure of the quality of the generated entangled states and is significant since applications depend critically on a minimum fidelity to be actually useful (e.g., quantum key distribution demands fidelity above 0.85, whereas distributed quantum computing needs fidelity beyond 0.9, approximately). Therefore, it is of interest to be able to accurately quantify both the entanglement rate and the fidelity in complex networks, at least to validate whether a particular quantum information processing task is possible.