Error Reconciliation Protocols for Continuous-Variable Quantum Key Distribution

Continuous-variable quantum key distribution (CV-QKD) utilizes an ensemble of coherent states of light to distribute secret encryption keys between two parties. One of the key challenges is the requirement of capacity-approaching error correcting codes in the low signal-to-noise (SNR) regime (SNR < 0 dB).

Multi-level coding (MLC) combined with multi-stage decoding (MSD) can solve this challenge in combination with multi-edge-type low-density parity-check (MET-LDPC) codes which are ideal for low code rates in the low SNR regime due to degree-one variable nodes. However, designing such highly efficient codes remains an open issue.

Here, we introduce the concept of generalized extrinsic-information transfer (G-EXIT) charts for METLDPC codes and demonstrate how this tool can be used to analyze their convergence behavior. We calculate the capacity for each level in the MLC-MSD scheme and use G-EXIT charts to exemplary find codes for some given rates which provide a better decoding threshold compared to previously reported codes.

In comparison to the traditional density evolution method, G-EXIT charts offer a simple and fast asymptotic analysis tool for MET-LDPC codes.A linear optimization approach to design highly efficient MET-LDPC codes at very low SNR, which is highly required by certain applications like CV-QKD will be discussed.

The cascade structure is introduced in terms of three disjoint submatrices and a convex optimization problem is proposed to design highly efficient MET-LDPC codes based on cascade structure. Simulation results show that the proposed algorithm is able to design MET-LDPC codes with efficiency higher than 95%, especially at very low SNR.