Faithful long-distance quantum information transmission is crucial for advancements in secure communication, sensing networks, and distributed quantum computing. While significant progress has been made in demonstrating entanglement distribution over long distances, intercontinental quantum communication remains challenging due to signal attenuation and degradation caused by loss and operational errors. Two main quantum repeater architectures exist: two-way repeaters, requiring long-lived multi-mode quantum memories, and one-way repeaters, employing quantum error-correcting codes to combat both loss and operational errors. One-way repeaters offer high communication rates determined by local processing times; however, they often demand substantial resources. Previous approaches, such as using only loss-tolerant photonic tree-cluster encoding, significantly reduce resource requirements but lack fault tolerance against operational errors, requiring extremely low operational error rates for long-distance transmission. Concatenation of different quantum error-correction codes, such as continuous-variable (CV) Gottesman-Kitaev-Preskill (GKP) codes with small discrete-variable (DV) codes, has been explored. However, generating high-quality optical GKP states remains experimentally challenging. This paper proposes a purely DV-based one-way quantum repeater architecture that uses code concatenation and flag-based quantum error correction to achieve fault tolerance in a resource-efficient manner, overcoming the limitations of previous approaches.

The paper reviews existing quantum repeater architectures, highlighting the trade-offs between two-way and one-way approaches. It discusses the limitations of previous one-way repeaters which often require a high number of qubits and complex operations. The authors also discuss the use of loss-tolerant codes such as the tree-cluster code and the challenges associated with continuous-variable codes such as Gottesman-Kitaev-Preskill codes. Previous work on concatenated codes to improve the efficiency of quantum repeaters are reviewed to put this work in context.

The proposed repeater architecture uses code concatenation, combining a loss-tolerant tree-cluster code as the inner code and a 5-qubit code as the outer code. The methodology involves three main steps: encoding, re-encoding and error correction, and decoding. **Encoding:** The message qubit is first encoded into the [[5, 1, 3]] 5-qubit code. Each of the five data qubits is then further encoded into a photonic tree-cluster state using a Bell state measurement. These five tree-encoded qubits are then transmitted in parallel. **Re-encoding and Error Correction:** The repeater network consists of two types of nodes: TYPE I nodes correct for loss using the tree code, and TYPE II nodes correct for both loss and operational errors using both the tree and 5-qubit codes. TYPE I nodes decode and re-encode the tree-cluster states, while TYPE II nodes perform fault-tolerant error correction on the 5-qubit code using flag-based stabilizer measurements and teleported CNOT gates. The model incorporates errors from re-encoding, noisy two-qubit gates, and potential erasure errors (loss of one or more tree-cluster states). The accumulated re-encoding error between TYPE II nodes (εtrans) is modeled using a single-qubit depolarizing channel. The erasure error correction is implemented by projecting the state back into the logical codespace of the 5-qubit code using a lookup table based on the measured syndromes. Syndrome extraction in TYPE II nodes is performed fault-tolerantly using a flag qubit. **Decoding:** At the end node, the five tree-cluster states are decoded, error correction based on the accumulated syndromes is applied, and the message qubit is recovered. The paper details a modular implementation using cavity-coupled quantum emitters and memory spins, with teleported gates used for non-local operations in TYPE II nodes. The performance is benchmarked using the secret key rate of a six-state Quantum Key Distribution (QKD) protocol, considering both fidelity and transmission probability. A cost function is minimized to optimize the repeater network configuration (number and type of nodes, inter-node distance, tree parameters) for different error rates and relative costs of TYPE I and TYPE II nodes.

The key findings demonstrate that the proposed concatenated code repeater significantly improves performance compared to a homogeneous repeater using only the tree code. The concatenated scheme achieves significantly higher secret key rates over long distances (thousands of kilometers) even with relatively high re-encoding error probabilities (~0.1%). The optimization of the repeater architecture, considering the relative costs of different node types, shows that a hybrid architecture with more TYPE I nodes and fewer, more complex TYPE II nodes can achieve near-optimal performance with a reduced resource cost. For example, at 1000 km and a re-encoding error rate of 0.1%, the concatenated repeater achieves a secret key rate of approximately 5.5 kHz, while the homogeneous repeater achieves less than 1 Hz. The optimized inter-repeater distances are shown to decrease with increasing re-encoding error rates to mitigate the accumulated errors. The ratio of TYPE I to TYPE II nodes varies depending on the total distance and the relative costs of the node types, showcasing the architecture's adaptability to different error scenarios and resource constraints. A modular implementation with few-qubit processors consisting of a cavity-coupled emitter and a small number of memory spins is proposed, making it suitable for implementation with current technology.

The results address the research question of creating a resource-efficient fault-tolerant quantum repeater by demonstrating the effectiveness of the proposed code concatenation scheme. The significance lies in the substantial improvement in long-distance communication performance with significantly higher secret key rates compared to previous non-fault-tolerant approaches. This is achieved while minimizing the resource overhead, making the proposal experimentally feasible. The adaptability of the architecture through optimization, considering varying error rates and relative node costs, highlights its robustness and practical applicability. The modular design based on current hardware capabilities further enhances its practicality.

The paper successfully demonstrates a resource-efficient and fault-tolerant one-way quantum repeater architecture using code concatenation. The hybrid approach, combining loss-tolerant and operational-error correcting codes, significantly improves long-distance communication performance. The modular design and optimization strategies presented make the proposed architecture a promising candidate for near-future experimental implementations. Future research could investigate the use of different quantum error-correcting codes and explore more sophisticated error models to further enhance performance and resource efficiency.

The model makes simplifying assumptions, such as neglecting correlated errors in the tree-cluster state generation and assuming efficient frequency conversion to the telecom band. The optimization is performed under specific constraints, and the results may vary with different parameter choices. The analysis focuses on a particular QKD protocol, and the performance may differ with other protocols. Furthermore, the assumption that decoherence errors in memory spins are negligible could be a limitation, depending on the actual implementation and hardware.