Fault-tolerant measurement-device-independent quantum key distribution with noisy non-Gaussian error correction
It is well known that the repeater node is an essential ingredient for the future global quantum network, which will enable high-rate private communication and entanglement distribution over very long distances. The near-term repeater architecture uses the measurement-based node that operate without both entanglement and quantum memory, which is the main idea of the measurement-device-independent quantum key distribution (MDI-QKD) protocol. The MDI-QKD protocol removes the trust condition from the inter repeaters, while its continuous variable (CV) version, when proposed, benefited from its deterministic nature, compatible with the classical devices, and shows a high rate for the short-range local area network (LAN). Whilst the theoretical backbone of CV-MDI-QKD protocol is well established, its secure transmission range is yet limited for practical LAN. In this study, we propose an enhanced scheme for the asymmetric CV-MDI-QKD protocol by using Gottesman-Kitaev-Preskill (GKP) oscillators-to-oscillators codes, where both loss error and operation error are suppressed to below the break-even point, without any delays caused by classical heralding signals. In particular, the proposed scheme, which correlates the noises of the data and ancilla via a pair of symplectic transforms, extracts the error syndromes from stabilizer measurements on the ancilla mode and informs the data mode for a corrective displacement. Numerical analysis shows the composable finite-size security of the protocol under the collective Gaussian attack, encompassing noiseless and noisy GKP states, with both wired (i.e., fiber-based) and wireless (i.e., free-space) configurations. In addition, we demonstrate that the residual errors of the GKP code can be further reduced by the concatenation method but has a trade-off between the layers number and the finite GKP squeezing.