Quantum Error Correction for Hybrid Quantum Repeater

We discuss quantum error correction (QEC) protocols for quantum repeaters based upon atomic qubit-entanglement distribution through optical coherent-state communication (hybrid quantum repeater). The effect of photon losses on a coherent-state qubit is a decrease of the coherent-state amplitude and a random phase flip error; the latter effect leads to the distribution of imperfect entanglement between neighboring repeater stations, when the light field is sent through the lossy communication channel and subsequently measured for conditional two-qubit entangled-state preparation. The conceptually simplest approach is then to perform entanglement distillation and swapping on the level of the qubit states in order to enhance the fidelity and increase the distance of the distributed entangled states. However, this approach requires complicated local quantum logic. Here we consider alternative ways to suppress the effect of photon losses for entanglement distribution. In particular, QEC codes can be applied to the optical mode that mediates the interaction between the qubits. An example for a possible QEC scheme is similar to the well-known 3-qubit phase flip code. We describe a scheme in which this code is employed for entanglement distribution and compare its performance with the canonical schemes based upon entanglement distillation. The encoding and decoding are based on dispersive light-matter interactions and homodyne detection can be used for syndrome measurement. Thus, only feasible elements are required with no need for photon-number resolving detectors, non-Gaussian optical gates, or quantum gate teleportation.