Quantum repeaters are critical in the development of the quantum internet because they enable quantum communication over long distances. Third-generation quantum repeaters or one-way quantum repeaters are the most advanced quantum repeaters that do not require two-way com- munication between nodes. Borregaard et al. (2019) proposed an architecture of one-way quantum repeaters based on photonic tree-cluster states. Tree-cluster states are highly entangled multi-qubit states that can protect quantum information from transmission losses. Creating these highly entangled multi-qubit states is a challenging and error-prone process. Conventional tree-cluster generation methods use an emitter that emits and entangles with photons. These methods, however, suffer from multiple photon emissions, which leads to errors. This paper presents a system for generating tree-cluster states that employ two different spin-cavity systems. The first spin-cavity system generates single photons using a cavity-assisted Raman scheme that prevents multiple photons from being emitted. The generated photons are scattered with a phase dependent on the spin state by the other spin-cavity system. The analytical model of the two spin-cavity systems for producing and scattering single-photons is presented. Furthermore, we model the errors that may occur during these processes.
We introduce a protocol for implementing a CZ gate between photons and spin. After that, this protocol is used to generate tree-cluster states. Furthermore, we optimize the entanglement between spin and photons by using the detuning between the two spin-cavity systems. Finally, we generate tree-cluster states using the spin-cavity system model and the photon-spin entanglement protocol. Following that, the effect of imperfect entanglement on the fidelity of the generated tree-cluster states is investigated. The fidelity of tree-cluster states with imperfect entanglement is then compared to the fidelity of tree-cluster states with single-qubit depolarising errors on photons. Unfortunately, we could not determine the relation between imperfect entanglement and the fidelity of the generated tree clusters in this study. Furthermore, we did not investigate the effects of imperfect entanglement on the encoding and decoding of information in tree-cluster states.

The generation of multiple entangled qubit pairs between distributed nodes is a prerequisite for a future quantum Internet. To achieve a practicable generation rate, standard protocols based on photonic qubits require multiple long-term quantum memories, which remains a significant experimental challenge. In this paper, we propose a novel protocol based on 2m-dimensional time-bin photonic qudits that allows for the simultaneous generation of multiple (m) entangled pairs between two distributed qubit registers and we outline a specific implementation of the protocol based on cavity-mediated spin-photon interactions. By adopting the qudit protocol, the required qubit memory time is independent of the transmission loss between the nodes, in contrast to standard qubit approaches. As such, our protocol can significantly boost the performance of near-term quantum networks.

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