Quantum key distribution over quantum repeaters with encoding

Quantum networks allow for the transmission of quantum information between physically separated quantum processors and can be used for both quantum communications and quantum computation applications. An enabling technology for future quantum networks is that of quantum repeaters (QRs). In this thesis, we study the performance of a quantum key distribution (QKD) system that is run over QRs with encoding. In such repeaters, quantum error correction techniques are used for entanglement distillation. We develop reliable and efficient tools, based on the linearity and transversality properties of the system, to obtain and study the shared states between two end users via such a repeater chain. We propose a post-selection technique which relies on the error-detection, rather than the error-correction, capability of the underlying code to sift out cases where an error has been detected. This simple but effective approach not only considerably improves the secret key rate and increases the resilience of the system to errors, but also simplifies the demonstration of such protocols in the near future. In this thesis, we mainly implement our techniques for three- and five-qubit repetition codes by modeling different resources of error in crucial components of the system. By developing several scalable numerical and analytical techniques, we investigate in detail the resilience of the setup to those imperfections in gates, measurement modules, and the initialization of the setup, at any nesting levels we are interested in. Furthermore, we propose two alternative decoder structures for encoded repeaters that not only boost system performance but also make the implementation aspects easier by removing two-qubit gates from the QKD decoder. We compare this class of QRs against alternative fully probabilistic settings and benchmark the regimes of operation, where one class of repeater outperforms the other. We find that there are feasible regimes where encoded repeaters—based on simple three-qubit repetition codes—could offer practical advantages. In order to get a view of how this type of QRs may behave in real life, among various promising candidates nowadays which enable deterministic entanglement swapping and distillation operations, here, we particularly investigate the suitability of platforms using nitrogen-vacancy (NV) centers in diamond as quantum memories. NV centers offer a two-qubit register, corresponding to their electron and nuclear spins, which makes it possible to perform deterministic two-qubit operations within one NV center. For QR applications, we however need to do joint operations on two separate NV centers. In this thesis, we study two NV-based repeater structures that enable such deterministic joint operations. One structure offers less consumption of classical communication, hence is more resilient to decoherence effects, whereas the other one relies on fewer numbers of physical resources and operations. We assess and compare their performance for the task of secret key generation under the influence of noise and decoherence with current and near-term experimental parameters. We quantify the regimes of operation, where one structure outperforms the other, and find the regions where encoded QRs offer practical advantages over their non-encoded counterparts