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

## TL;DR

This paper presents a GPU-accelerated quasi-cyclic LDPC code decoding method for long-distance continuous-variable quantum key distribution, significantly extending the maximum distance and throughput compared to previous approaches.

## Contribution

Introduction of a GPU-optimized quasi-cyclic LDPC code construction for efficient key reconciliation in CV-QKD, enabling longer distances and higher throughput.

## Key findings

- Achieved 160km maximum distance in CV-QKD with high throughput
- Decoding throughput of 1.72 Mbit/s on GPU for 10^6 bit block length
- Extended previous maximum distance from 100km to 160km

## Abstract

The speed at which two remote parties can exchange secret keys over a fixed-length fiber-optic cable in continuous-variable quantum key distribution (CV-QKD) is currently limited by the computational complexity of post-processing algorithms for key reconciliation. Multi-edge low-density parity-check (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 10^6 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 4.10x10^{-7} bits/pulse for a rate 0.02 multi-edge code with block length of 10^6 bits when finite-size effects are considered. This work extends the previous maximum CV-QKD distance of 100km to 160km, while delivering between 1.07x and 8.03x higher decoded information throughput over the upper bound on the secret key rate for a lossy channel. The GPU-based QC-LDPC decoder achieves a 1.29x improvement in throughput over the best existing GPU decoder implementation for a rate 1/10 multi-edge LDPC code with block length of 2^{20} bits. These results show that LDPC decoding is no longer the computational bottleneck in long-distance CV-QKD.