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Physical Review A

High-efficiency quantum-key-distribution error-correction algorithm based on the Gilbert-Elliott model and the low-density parity-check code

Ke Cui, Xianglong Xia, Simeng Li, Ming Bo Zhu, Jian Sun, Shibiao Tang

Published Aug 11, 2025• 0 citations
Publisher

Abstract

For the post-processing of the quantum-key-distribution (QKD) system, the error correction is the most important step, whose efficiency significantly influences the final key generation rate. Traditionally, the error-correction algorithm is developed based on the binary symmetrical channel model. However, this may not comply with the actual channel status. For example, for the fiber-based QKD system, differentiable statuses of the fiber (unstable or steady) should exist for some time and cause different bit error rates. If this channel feature can be exploited, it may considerably improve the error-correction efficiency. This paper proposes to introduce the Gilbert-Elliott (GE) model to analyze the QKD channel. The error-correction algorithm adopts the low-density parity-check (LDPC) code. The GE nodes are added into the traditional belief propagation (BP) algorithm of the LDPC decoder. Simulation results showed that the remaining frame error rate could be improved from ${10}^{\ensuremath{-}2}$ (or ${10}^{\ensuremath{-}3}$) to ${10}^{\ensuremath{-}4}$. An actual QKD system with the quantum channel state under manual control was adopted to collect the key bit string. By analyzing the occurrence probability of the $k$ consecutive "1," the key bit string showed closely matched distribution with the GE model prediction. After the error correction, the remaining bit error rate of the modified BP algorithm was significantly smaller than that of the standard BP algorithm with an improvement of more than 2 orders of magnitude. Both simulation and actual experimental results demonstrate that our proposed method can greatly improve the error-correction efficiency if the channel characteristic of the QKD system is accurately exploited, which provides an efficient way to generate a higher key rate.

DOI
10.1103/xnb7-cghn

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For the post-processing of the quantum-key-distribution (QKD) system, the error correction is the most important step, whose efficiency significantly influences the final key generation rate. Traditionally, the error-correction algorithm is developed based on the binary symmetrical channel model. However, this may not comply with the actual channel status. For example, for the fiber-based QKD system, differentiable statuses of the fiber (unstable or steady) should exist for some time and cause different bit error rates. If this channel feature can be exploited, it may considerably improve the error-correction efficiency. This paper proposes to introduce the Gilbert-Elliott (GE) model to analyze the QKD channel. The error-correction algorithm adopts the low-density parity-check (LDPC) code. The GE nodes are added into the traditional belief propagation (BP) algorithm of the LDPC decoder. Simulation results showed that the remaining frame error rate could be improved from 10\ensuremath−2{10}^{\ensuremath{-}2}10\ensuremath−2 (or 10\ensuremath−3{10}^{\ensuremath{-}3}10\ensuremath−3) to 10\ensuremath−4{10}^{\ensuremath{-}4}10\ensuremath−4. An actual QKD system with the quantum channel state under manual control was adopted to collect the key bit string. By analyzing the occurrence probability of the kkk consecutive "1," the key bit string showed closely matched distribution with the GE model prediction. After the error correction, the remaining bit error rate of the modified BP algorithm was significantly smaller than that of the standard BP algorithm with an improvement of more than 2 orders of magnitude. Both simulation and actual experimental results demonstrate that our proposed method can greatly improve the error-correction efficiency if the channel characteristic of the QKD system is accurately exploited, which provides an efficient way to generate a higher key rate.
Quantum encryption guarantees the absolute security of key distribution from the physical mechanism. However, in the process of quantum key distribution (QKD), quantum channel noise will reduce the transmission efficiency. Quantum low-density parity check (quantum LDPC) code has become a hot research topic of quantum error correcting code due to its great flexibility in the choice of code length and code rate and its dependence on sparse graphs. In order to improve the transmission reliability, a QKD protocol based on BB84 and quantum LDPC code is designed. First, we construct a quantum LDCP error correcting code. Then we utilize the error correcting code and BB84 to construct our QKD protocol. Finally, we analyze the influence of the quantum LDPC code on the proposed protocol. The results show that the key transmission efficiency of the our protocol is improved, and the validity of the QKD protocol based on BB84 and quantum LDPC codes in noisy quantum channels is verified.
In this paper, we study the performance of different graph-based error-correcting codes over Gilbert-Elliott (GE) channels. We propose a hybrid coding scheme in which each code bit is checked by both a parity-check code and a Hamming code. A hybrid code can be represented by a code-to-code graph, which can be used to optimize the code. Asymptotic minimum distance properties of the hybrid code are derived, and it is shown that the expected minimum distance of the hybrid code increases linearly with respect to the code length. Simulation results show that for a typical GE channel, hybrid codes can outperform regular low-density parity-check codes by more than an order of magnitude, in terms of bit-error rate.
In this paper, we investigate the design of low-density parity-check (LDPC) codes for the Gilbert-Elliott (GE) channel. Recently, Eckford et al. proposed a design method of irregular LDPC codes using approximate density-evolution (DE) for Markov channels. In the design method proposed by Eckford et al., the probability density function (PDF) of the messages from variable nodes to check nodes is approximated by the Gaussian distribution. In this paper, we first show the method to obtain the accurate PDF of the messages from variable nodes to check nodes by utilizing two DE steps for the Gaussian distribution. We call this method the iterative density approximation (IDA). Using this method, we can design the good LDPC codes. Next, we propose an efficient design method of irregular LDPC codes by using Beta approximation to the PDF of the channel state probability for the GE channel. Consequently, we show that the complexity to calculate PDFs of the channel messages is considerably reduced though the rates of LDPC codes obtained by using the proposed approximation are almost the same as that of the IDA method.
The problem of low-density parity-check (LDPC) decoding in channels with memory has been attracting increasing attention in the literature. In this thesis, we use LDPC codes in an estimation-decoding scheme for the Gilbert-Elliott (GE) channel and more general Markov-modulated channels. The major accomplishments of this thesis include both analysis and design components. To analyze our estimation-decoding scheme, we derive density evolution for the GE channel, which has previously been used largely in memoryless channels. Furthermore, we develop techniques that use density evolution results to more efficiently characterize the space of Markov-modulated parameters. We begin by applying a characterization to the GE channel, following which we generalize this characterization into a partial ordering of Markov-modulated channels in terms of probability of symbol error. We also consider the design problem of developing LDPC degree sequences that are optimized for the GE channel. We obtain a novel design tool that approximates density evolution for our estimation-decoding algorithm, and present degree sequences that represent the best known codes in the GE channel. We also present a method of generalizing this tool to Markov-modulated channels, and give some of the first optimized degree sequences ever obtained for these channels.
Abstract Error correction is a significant step in postprocessing of continuous-variable quantum key distribution system, which is used to make two distant legitimate parties share identical corrected keys. We propose an experiment demonstration of high speed error correction with multi-edge type low-density parity check (MET-LDPC) codes based on graphic processing unit (GPU). GPU supports to calculate the messages of MET-LDPC codes simultaneously and decode multiple codewords in parallel. We optimize the memory structure of parity check matrix and the belief propagation decoding algorithm to reduce computational complexity. Our results show that GPU-based decoding algorithm greatly improves the error correction speed. For the three typical code rate, i.e., 0.1, 0.05 and 0.02, when the block length is 10 6 and the iteration number are 100, 150 and 200, the average error correction speed can be respectively achieved to 30.39 Mbits/s (over three times faster than previous demonstrations), 21.23 Mbits/s and 16.41 Mbits/s with 64 codewords decoding in parallel, which supports high-speed real-time continuous-variable quantum key distribution system.
For high-speed quantum key distribution (QKD) post processing, information reconciliation is the most computational step, which usually acts as the system speed bottleneck. Reconciliation efficiency and operation throughput are the two most important performance parameters, but they are often compromised to each other in actual realization due to the available hardware resources. Another characteristic of the information reconciliation is that its channel is time-varying, and in order to guarantee high efficiency, some rate-compatible error correction scheme should be adopted. To cope with the above problems, this paper proposes a rate-compatible high-efficiency reconciliation algorithm based on quasi-cyclic (QC) low-density parity-check (LDPC) codes and puncturing algorithms. On the other hand, in order to obtain high throughput while maintaining low implementation complexity, the normalized min-sum algorithm (NMSA) is realized and optimized by using the fast prototyping tool of high-level synthesis (HLS). The proposed information reconciliation module was implemented on the Xilinx Zynq Ultrascale+ ZCU102 development board. Experimental results show that the maximum throughput rate of the implemented reconciliation module in this paper can reach 136 Mbps, while the efficiency factor is kept lower than 1.32 across the error rate range of 1.7%∼10.6% under the remaining frame error rate level of 10−3. The comprehensive good performance obtained by our design can strongly support the development of modern high-speed QKD systems.