Key Reconciliation with Low-Density Parity-Check Codes for Long-Distance Quantum Cryptography

The speed at which two remote parties can exchange secret keys over a fixed-length fiber-optic cable in continuousvariable quantum key distribution (CV-QKD) is currently limited by the computational complexity of post-processing algorithms for key reconciliation. Multi-edge low-density paritycheck (LDPC) codes with low code rates and long block lengths were proposed for CV-QKD, in order to extend the maximum reconciliation distance between the two remote parties. Key reconciliation over multiple dimensions has been shown to further improve the error-correction performance of multi-edge LDPC codes in CV-QKD, thereby increasing both the secret key rate and distance. However, the computational complexity of LDPC decoding for long block lengths on the order of 10 bits remains a challenge. This work introduces a quasi-cyclic (QC) code construction for multi-edge LDPC codes that is highly suitable for hardware-accelerated decoding on a modern graphics processing unit (GPU). When combined with an 8-dimensional reconciliation scheme, the LDPC decoder achieves a raw decoding throughput of 1.72Mbit/s and an information throughput of 7.16Kbit/s using an NVIDIA GeForce GTX 1080 GPU at a maximum distance of 160km with a secret key rate of 4.10×10−7 bits/pulse for a rate 0.02 multi-edge code with block length of 10 bits when finite-size effects are considered. This work extends the previous maximum CV-QKD distance of 100km to 160km, while delivering between 1.07× and 8.03× higher decoded information throughput over the upper bound on the secret key rate for a lossy channel. For distances beyond 130km, the GPU decoder delivers an information throughput between 1868× and 18790× higher than the achievable secret key rates with a 1MHz light source. The GPU-based QC-LDPC decoder achieves a 1.29× improvement in throughput over the best existing GPU decoder implementation for a rate 1/10 multi-edge LDPC code with block length of 2 bits. These results show that LDPC decoding is no longer the computational bottleneck in long-distance CV-QKD, and that the secret key rate remains limited only by the physical parameters of the quantum channel and the latency of privacy amplification. Keywords—Quantum key distribution, QKD, continuous-variable QKD, reverse reconciliation, LDPC codes, LDPC decoding, multiedge LDPC codes, quasi-cyclic LDPC codes, GPU decoding.