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Information reconciliation of continuousvariables quantum key distribution: principles, implementations and applications
EPJ Quantum Technology volume 10, Article number: 40 (2023)
Abstract
Quantum key distribution (QKD) can provide informationtheoretically secure keys for two parties of legitimate communication, and information reconciliation, as an indispensable component of QKD systems, can correct errors present in raw keys based on errorcorrecting codes. In this paper, we first describe the basic knowledge of information reconciliation and its impact on continuous variable QKD. Then we introduce the information schemes and the corresponding error correction codes employed. Next, we introduce the ratecompatible codes, the hardware acceleration of the reconciliation algorithm, the research progress of information reconciliation, and its application in continuous variable QKD. Finally, we discuss the future challenges and conclude.
1 Introduction
Quantum key distribution (QKD) [1–9] allows legitimate parties, Alice and Bob, to share secure keys through an insecure quantum channel. The fundamental theorems of the quantum physics guarantee that nonorthogonal quantum states transmit through a quantum channel cannot be replicated accurately. Furthermore, any measurements trying to discriminate the nonorthogonal quantum states will inevitably disturb them. Therefore, any eavesdropping behaviors against on QKD can be discovered.
According to the different carriers of the key, QKD can be divided into discrete variable QKD (DVQKD) and continuous variable QKD (CVQKD) [10–14]. DVQKD uses the polarization or phase of single photons to encode the key information, which can realize longdistance key distribution by using single photon detection technology. CVQKD employs the quadrature components of quantum states to encode the key information, it is compatible with the existing coherent optical communication technology and can achieve high key rate in short and medium distance.
The first concept of CVQKD was proposed in 1999 [15]. The Gaussianmodulated coherent states protocol was proposed in 2002 (GG02 protocol) [16] and experimentally verified in 2003 [17]. So far, a series of important advances have been reported in protocol design, experiment implementation, security analysis and field test. Novel theoretical protocols [18–23] are designed, security proofs are constantly improving [24–31], experiments [32–40] are gradually moving from proofofprinciple lab demonstrations to prototypes and infield implementations. The transmission distance and the secret key rate (SKR) of systems continue to improve. The pointtopoint transmission reaches 202.81 km of ultralowloss optical fiber [33]. Measurementdeviceindependent CVQKD has been successfully verified [19, 37]. Recently, some research teams have tried to integrate the sender and receiver of CVQKD on silicon photonic chips [41–44]. Furthermore, the feasibility of establishing secure satellitetoground CVQKD links is investigated [45, 46].
A typical CVQKD system usually consists of four parts [11]: (1) Preparation, distribution and measurement of quantum states; (2) Key sifting and parameter estimation; (3) Information reconciliation (IR); and (4) Privacy amplification (PA). In the IR stage, Alice and Bob obtain the identical bit strings by correcting errors between their raw keys. As an indispensable step in CVQKD system, IR must be able to match or surpass the clock rate of the system. However, the IR of CVQKD is relatively complicated because it usually works at a low signaltonoise ratio (SNR) regime, and digital signal processing technologies sometimes need to be introduced to improve the SNR [23, 47]. An overview of IR in DVQKD has been presented in [48]. In this paper, we review the IR of CVQKD including its principles, implementations, and applications.
The rest of the paper is organized as follows. In Sect. 2, we provide basic knowledge of IR and discuss the impact of IR performance on the CVQKD system. It follows a summary of relevant reconciliation protocols, including slice reconciliation, multidimensional reconciliation, and other improved protocols in Sect. 3. In Sect. 4, we review the error correction codes (ECCs) that are used in IR, including lowdensity paritycheck (LDPC) codes, polar codes, Raptor codes, and spinal codes. Section 5 presents the improvement of IR throughput based on hardware, such as fieldprogrammable gate array (FPGA) or graphics processing unit (GPU). Next, we present the research progress of IR in Sect. 6. The typical applications of IR in CVQKD systems are discussed in Sect. 7. In Sect. 8, we discuss the current challenges of IR. Finally, we give a conclusion in Sect. 9.
2 Preliminaries
In QKD, inconsistencies inevitably exist in the raw keys obtained by communication parties due to the noises and attenuation in quantum channel and the noises of quantum states themselves. The aim of IR is to share a set of completely consistent key bits by using classical ECCs for Alice and Bob.
2.1 Performance parameters of IR
The two most critical parameters of QKD are SKR and transmission distance, respectively. The multiple parameters of IR significantly affect the performance of QKD. Table 1 lists the three key parameters used for evaluating IR’s performance. They are reconciliation efficiency (β), frame error rate (\(FER\)), and throughout (T).
The reconciliation efficiency β is the most important parameter for evaluating the quality of a key reconciliation scheme. Reconciliation efficiency is used to characterize the efficiency of the error correction. Throughput T tells the number of raw keys processed per unit of time. Notice that fast IR with throughput no less than the system clock rate is the prerequisite of a realtime QKD. Hardware acceleration, such as FPGA or GPU, can be used to improve the throughput. When ECCs are used for IR, they are carried out in the unit of code blocks, which is called a “frame”. The raw keys usually need to be divided into a series of frame and decoded. The FER represents the failure probability of IR. Obviously, the lower the FER, the better.
2.2 Effects of IR on CVQKD systems
Ideally, the asymptotic SKR of CVQKD systems can be expressed as:
where \({{I}_{AB}}\) is the Shannon mutual information between Alice and Bob, \({{\chi }_{BE}}\) is the Holevo bound.
Considering the realistic reconciliation efficiency, FER, and throughput, the SKR of the CVQKD system can be expressed as:
where \(\gamma =P{{P}_{out}}/P{{P}_{in}}\), \(PP_{out}\) and \(PP_{in}\) represent the postprocessing (include IR and PA) output and input rates, respectively. The value of \(PP_{out}\) determines the maximum QKD clock rate that the postprocessing can support and γ satisfies \(0\le \gamma \le 1\). When the speed of the postprocessing is greater than or equal to the generation speed of the raw keys, \(\gamma = 1\), that is, the raw keys can be fully utilized. The real throughput of IR will change the value of \(\gamma =P{{P}_{out}}/P{{P}_{in}}\), and ultimately affects the SKR.
Figure 1 shows the SKR as a function of the transmission distance (standard singlemode fiber channel with loss of 0.2 dB/km) at different reconciliation efficiencies (91%99%). As can be seen from Fig. 1, the higher the reconciliation efficiency, the farther the transmission distance. Given the transmission distance, higher reconciliation efficiency enables higher key rate.
When further consider the finite raw keys and consumed raw keys for parameter estimation, the practical secret key rate of the QKD system is given by [49]:
where n is the number of raw keys used to distill the secret key. N is the number of sifted raw keys after quantum transmission and measurement. \(\Delta (n)\) is the finitesize offset factor.
2.3 Direct and reverse reconciliation
A CVQKD system can be implemented in direct reconciliation (DR) or reverse reconciliation (RR), which have different performances. For DR, Alice sends the redundant information required for error correction to Bob, so that Bob can correct the errors of his data using the received information and obtain a bit string that is exactly the same as that of Alice. The ideal key rate of DR can be expressed as \(I_{AB}\chi _{AE}\). However, when the transmittance of the quantum channel is less than 1/2, it results in the inability to generate a secure key. For example, Eve can simulate a loss channel with a beam splitter and split the light emitted by Alice into two beams, then she keeping the larger part and sending the smaller part to Bob. In this case, Eve being able to extract more key information than Bob, which hinders legitimate parties to generate a secure key, called the 3 dB loss limit [50].
Two ways have been proposed to overcome the 3 dB limit, namely RR and postselection, respectively. For RR, Bob’s raw keys is used as the benchmark, and he sends the information required for error correction to Alice, who correct her bit string to be the same as Bob’s bit string. The ideal SKR of RR can be expressed as \(I_{AB}\chi _{BE}\). RR can dramatically extend the transmission distance and generate security keys over longer distances, thus becoming a dominant scheme.
3 Reconciliation schemes
In CVQKD systems, Alice and Bob obtain a set of correlated raw keys X and Y after the quantum states preparation, measurement, and key sifting phases. For Guassian modulated CVQKD protocols, the raw keys are Gaussian variables.
Several schemes have been proposed for IR of Gaussian symbols, such as slice reconciliation [51], multidimensional reconciliation [52], sign reconciliation [50], and so on [53, 54], each scheme covers a certain range of SNRs. As shown in Fig. 2, the slice reconciliation is suitable for relatively high SNR of larger than 1 (short transmission distance), and the multidimensional reconciliation is suitable for low SNR from 0.01 to 1 (long transmission distance).
3.1 Slice reconciliation
The slice reconciliation was first proposed in 2004 [51]. It can correct the errors of Gaussian symbols using binary ECCs. For reverse reconciliation, Bob uses the quantizing function \(Q:\mathbb{R}\to {{ \{ 0,1 \}}^{m}}\) to transform each Gaussian variable \({{Y}_{i}}\) into an mbit label \(\{ {{B}_{j}}({{Y}_{i}}) \},j=1,\ldots,m\). Next, Bob uses multilevel encoding (MLE) that encodes each individual level j of the label bits independently as the syndrome of an error correcting code with rate \({{R}_{j}} ( 1\le j\le m )\). To recover Bob’s mbit label \(\{ {{B}_{j}} \}\), Alice employs multistage decoding (MSD) and uses her own source X as side information. Finally, the two parties share identical keys. The principle of slice reconciliation is shown in Fig. 3(a).
To transform a Gaussian variable into a binary sequence, the real number axis is divided into a number of intervals and then a proper mapping is performed for the sliced Gaussian variables as shown in Fig. 3(c). The quantization efficiency of the Bob’s Gaussian variables can be expressed as:
where \(Q(Y)\) is the quantified Gaussian variable, and \(I(X;Y)\) is the mutual information between Alice and Bob. The quantization efficiency versus SNR for the optimal interval quantization and equal interval quantization is shown in Fig. 3(b).
The mutual information \(I(X;Q(Y))\) can be expressed as:
The terms on the rightside of Eq. (5) are defined as follows:
where \(t_{a1}\) and \(t_{a}\) denote the left and right endpoints of the interval a.
In Fig. 3(c), the raw key Y are quantified in to five levels. The information included in the first three levels is very small and disclosed directly, and the latter two levels are encoded and successively decoded. More precisely, Bob sends the quantized bit strings \(\{ {{B}_{1}}({{Y}_{i}}) \}\), \(\{{{B}_{2}}({{Y}_{i}}) \}\), and \(\{{{B}_{3}}({{Y}_{i}}) \}\) of the first three levels and the syndromes \({{S}_{4}}\) and \({{S}_{5}}\) of the last two levels to Alice.
The reconciliation efficiency β of slice reconciliation is given by
where \(H [ Q ( Y ) ]\) is the information entropy of \(Q ( Y )\).
Bai et al. [56] and Mani et al. [58] explored the quantification scheme and analyzed the 4levels, 5levels, and 6levels quantification and designed ECCs with better decoding performance to improve reconciliation efficiency. Wen et al. [59] proposed an improved slice reconciliation protocol, named RotatedSEC, which performs a random orthogonal rotation on the raw keys before quantization, and deduces a new estimator for the quantized sequences.
3.2 Multidimensional reconciliation
For CVQKD protocols, the raw keys of Alice and Bob are correlated Gaussian variables and the SNR will be very low for long transmission distances. In this case, the raw keys have a small absolute value and are distributed around 0. Thus, it is difficult to discriminate the sign and realize the encoding and decoding. The multidimensional reconciliation algorithm provides a powerful encoding scheme for low SNR scenario and thus effectively extend the key distribution distance. By this way, the channel between Alice and Bob is converted into a virtual binary input additive white Gaussian noise (AWGN) channel and therefore efficient binary codes can be employed. The highest reconciliation efficiency achieves at \(d=8\) because the ratio is highest between the capacities of the 8dimensional channels and the binary input AWGN channel [60].
The basic principle of improving the discrimination in multidimensional reconciliation is the rotation of the raw keys, as shown in Fig. 4(a). Consider that Alice and Bob share a set of correlated Gaussian variables X and Y and \(x_{1}\) and \(x_{2}\) belong to X. Figure 4(a) shows the four possible states that Bob needs to discriminate. (a1) represents the slice reconciliation: the four states are well separated, but the Gaussian symmetry is broken; (a2) represents the sign reconciliation: the symmetry is preserved but some states are difficult to discriminate because they are very close to each other; (a3) represents the multidimensional reconciliation, the states are well separated as well as the symmetry is preserved.
Both parties form a vector for each d elements of the Gaussian symbols, labeled \(\boldsymbol{X}'\) and \(\boldsymbol{Y}'\), and then normalize each ddimensional vector as follows:
where \(\vert \boldsymbol{X}' \vert \) and \(\vert \boldsymbol{Y}' \vert \) denote the modulo of vectors. After normalization, the random vectors are transformed into signal points on a unit sphere and the Gaussian variables are transformed into \(\boldsymbol{x}'\) and \(\boldsymbol{y}'\). Then, a set of random bit strings \(\boldsymbol{u}= \{ {{b}_{1}},{{b}_{2}},\ldots,{{b}_{i}} \}\) of the same length as the Gaussian variables that obeying a uniform distribution is generated by a true random number generator (TRNG) at the receiver side, and each random bit string is transformed into a ddimensional spherical vector as follows
The receiver performs a ddimensional spatial rotation operation to compute α, such that it satisfies \(\alpha \cdot \boldsymbol{y}' = \boldsymbol{u}'\). u is encoded to generate the syndrome S. Then α and S are sent to Alice. Alice uses the received information to compute the mapping function \(M'\). Then, she calculates v using the equation \({M}'\cdot \boldsymbol{x}' = \boldsymbol{v}\). Finally, Alice uses the computed v as side information to recover the exactly same u as Bob by decoding.
The reconciliation efficiency β of the multidimensional reconciliation is defined as:
where R is the rate of the ECC used in reconciliation scheme.
Recently, several works on optimization and modification of the multidimensional reconciliation and ECCs have been reported to further improve the performance of CVQKD systems.
In 2019, Li et al. proposed an initial decoding message computation method for multidimensional reconciliation [61], which does not need the norm information from the encoder. They show that the improved scheme can decrease the communication traffic and storage resource consumption without significantly degrading in the reconciliation efficiency. What is more, the improved scheme can decrease the secure key consumption for classical channel authentication.
In 2021, Feng et al. studied the SNR of the virtual channel of multidimensional reconciliation and proved that the noise of the virtual channel follows the Student’s tdistribution [62]. They proposed a novel tBP (beliefpropagation) decoding algorithm, whose FER is superior to the traditional BP decoding algorithm.
In 2023, Wang et al. applied the reverse multidimensional reconciliation to noGaussian modulation protocols [63], which can improve the performance of CVQKD. In their work, the variables X and Y are obtained after nonGaussian postselection [64] and X follows nonGaussian distribution and Y follows Gaussian distribution. Alice normalizes her nonGaussian distribution variables X and rotates her normalized raw keys through the mapping function.
Table 2 shows the comparisons of the slice and multidimensional reconciliation schemes.
3.3 Other schemes
Sign reconciliation [50] directly encode the continuous random variable to a key bit by using its sign. The sign reconciliation has the feature of simplicity and low complexity, however the performance is low. Jiang et al. proposed a new reconciliation scheme based on the punctured LDPC codes in 2017 [53]. Compared to the multidimensional reconciliation, their scheme has lower time complexity. Especially when the chosen punctured LDPC code achieves the Shannon capacity, there is no information leaked to the eavesdropper after IR stage. This indicates that the PA algorithm is no more required. Late [65], they proposed a new reconciliation scheme to decrease the FER. Gümüş et al. [54] proposed a multiple decoding attempts protocol that can be used in fourstate CVQKD and each attempt has fewer decoding iteration than the conventional protocol.
4 Error correction codes
ECCs [66] allow for the detection and correction of errors occurred during the data transmission. Since a CVQKD system usually works in a Gaussian channel with very low SNRs, typically below 0 dB, so that the initial bit error rate is very high, which requires that the ECC has very good decoding performance. Selecting an appropriate ECC with good decoding performance is the key to improve the reconciliation efficiency and reduce FER. A variety of ECCs with good decoding performance can be used to realize IR of CVQKD, such as LDPC codes, Polar codes, Raptor codes, Spinal codes and so on. Next, we will introduce these ECCs in detail.
4.1 LDPC codes
LDPC codes [66] are a good performance ECC with low decoding complexity and can be used to achieve highefficiency IR very close to the Shannon limit. They are usually represented using a sparse check matrix or a Tanner graph as shown in Fig. 5(a), in which edges are used to connect check nodes to variable nodes. Based on the information transfer between different variable nodes in the Tanner graph, LDPC codes use soft decision iterative decoding algorithm with posterior probability information. In addition, LDPC codes can be applied to almost all types of channels and parallel processing making them suitable for implementing hardware acceleration. LDPC codes are divided into different types, such as irregular LDPC codes [56, 57], multiedge type LDPC (METLDPC) code [58, 60], nonbinary LDPC codes [67], spatially coupled LDPC codes [65], and globally coupled LDPC code [68]. To achieve good performance, several aspects should be carefully considered, including decoding algorithms, message propagation schedule, and construction of the parity check matrix (PCM), etc.
4.1.1 Degree distribution
The degree distribution is used in irregular codes to describe the degree of the nodes, that is, the distribution of nonzero elements in the PCM. Assume that the maximum degree of the variable nodes and the check nodes are \(d_{v}\) and \(d_{c}\), respectively, the degree distributions of the variable and check nodes can be expressed by
where \(\lambda _{i}\) and \(\rho _{i}\) denotes the ratio of the number of edges connected to the variable or check nodes of the degree i to the total number of edges, respectively.
4.1.2 Construction of parity check matrix
In addition to the size of PCMs and the degree distributions of their nodes, the position of the edges has also a significant impact on the error correction performance. A number of approaches have been proposed, such as random construction, the progressive edge growth (PEG) algorithm, and quasicyclic (QC) codes. The previous work showed that the PEG algorithm has better performance at \(SNR \sim 3\), while random construction exhibits better performance at \(SNR \sim 1\) [56]. QC codes are defined by a PCM constructed from an array of \(q \times q\) cyclically shifted identity matrices and \(q \times q\) zero matrices. It imposes a highly regular PCM structure with a sufficient degree of randomness offset to achieve nearShannonlimit error correction performance while reducing the complexity of the decoder. It also reduces data permutation and memory access complexity by eliminating random, unordered memory access patterns.
4.1.3 Decoding algorithms
LDPC codes are usually decoded using a belief propagation (BP) algorithm, in which messages typically in the form of a logarithmiclikelihood ratio (LLR) are iteratively passed in both directions along the edges between connected nodes. Two dominant LDPC decoding algorithms are the sumproduct algorithm (SPA) and the minsum algorithm (MSA). Note that two modifications of the MSA, called the normalized MSA and offset MSA [69], have been proposed to improve the performance.
The message propagation schedule [70] of the LDPC decoding process determines the order in which variable nodes and check nodes are processed, as well as whether multiple nodes are processed in parallel. Flooding, layered belief propagation (LBP) [71], and informed dynamic scheduling are three widely used schedules. LBP tends to converge to the correct code word with fewer iterations and therefore has lower computational complexity, and the memory size required for a LBP decoder is half of that required for a flooding decoder.
Although MSAs are easy to implement on FPGA, the sumproduct decoding algorithm has better decoding performance where the layered message passing mechanism converges faster. It can reduce the number of iterations and the consumption of storage resources when applied to FPGAs [72].
The BP decoding algorithm consists of the following four main steps:
Step 1: Initialization: calculating the initial LLRs for each variable node:
Step 2: Check nodes information processing:
where l denotes the number of iterations.
Step 3: Variable nodes information processing:
Step 4: Decision. Computing harddecision information for all variable nodes:
If \(LLR_{i}^{(l)}\ge 0\), then \({{x}_{n}}=1\), otherwise \({{x}_{n}}=0\). If \(\boldsymbol{H} \cdot {{X}^{T}} = \boldsymbol{S}\), the result X is the decoded output, otherwise return to step 2. Repeat steps 2 and 3 until \(\boldsymbol{H} \cdot {{X}^{T}} = \boldsymbol{S}\) is satisfied, or the number of iterations reaches the given maximum value.
In the LBP algorithm, step 2 and step 3 will be combined together. More precisely, the nodes in the PCM are processed rowbyrow:
where the function Ψ is defined as \(\Psi ( x )=\log [ \tanh ( { \vert x \vert }/{2} ) ]\). From the node processing equation, the intermediate variables do not need to be cached into the next iteration process, thus effectively reducing the consumption of storage resources when implementing on FPGA.
4.2 METLDPC codes
Richardson and Urbanke proposed the METLDPC code [73]. By introducing new constraints into the code design, the METLDPC code has the following advantages: (1) it shows better error correction performance on Gaussian noise channels; (2) the error correction performance at low code rates is closer to the Shannon limit than other ECCs, which can reduce the error floor at very low SNRs; (3) it also has good error correction performance at high code rates.
In 2011, Jougue et al. [60] used density evolution algorithm to find a set of degree distributions with code rate of 0.02, and then applied METLDPC codes to multidimensional reconciliation. They achieved a reconciliation efficiency of 96.9% when the code rate was 0.02 at the SNR of 0.029. In 2017, Wang et al. [74] used density evolution to find degree distributions at code rates of 0.05 and 0.10. In 2021, Mani et al. [58] used generalized external message passing graph method to find the best degree distributions at code rates of 0.01, 0.02, 0.05, and 0.10, respectively. Table 3 summarizes the optimal node degree distributions at different code rates.
4.3 Polar codes
Polar codes are proposed by Arıkan in 2009 [78]. Arıkan proved that polar codes could achieve channel capacity in binary symmetric channels, and then Korada et al. showed that polar codes could also achieve channel capacity in arbitrary binary input discrete channels. Furthermore, the encoding/decoding of polar codes can be achieved with low complexity. Due to its good performance, it is applied to 5G communication [79, 80] and has also been applied to the IR of CVQKD in recent years [49, 59, 81–85].
The basic idea of polar codes is the polarized subchannels exhibiting different properties by channel polarization at the encoding side. When the code length increases to a certain level, the channel capacity of some subchannels converges to 1 while the channel capacity of other subchannels converges to 0. Then, the information can be transmitted on the subchannels whose channel capacity is close to 1 to approach the channel capacity as much as possible. However, the coding subchannels of the finite codelength polar codes are not entirely polarized, and some of them are neither completely noisefree nor completely noisy, called the intermediate channels. The intermediate channels are protected by outer LDPC codes, called ICLDPC Polar codes [84].
Since Arıkan proposed the basic decoding algorithms: polarization codessuccessive cancellation decoding algorithm, and belief propagation decoding algorithm [78], the decoding algorithms of polarization codes have been improved after more than ten years of efforts (for more details, please refer to Ref. [86]).
The main description tools of encoding and decoding for polar codes are “Trellis” and “Code Tree”, where the latter is a compressed form of the former. In general, given a polar code of code length \(N = 2n\), the trellis contains n levels. The schematic diagram of trellis is shown in Fig. 5(c). The left side is the source side and the right side is the channel side.
4.4 Ratecompatible codes
The ratecompatible codes are necessary for practical CVQKD systems to deal with the timevarying channels. The methods to achieve ratecompatible are nodes puncturing, nodes shortening, and rateless codes. Typical rateless codes are Luby Transform (LT) codes [87], Raptor codes [88], and spinal codes [89]. The nodes puncturing and shortening (P&S) method, Raptor codes, and spinal codes that have been used in IR are elaborated in the following section.
4.4.1 Puncturing and shortening
Puncturing [90] and shortening [91] are two ratecompatibility techniques that can increase and decrease the code rate, respectively. Using the two methods, we can achieve ratecompatible ECCs for channels with varying SNRs. P&S can be applied not only to LDPC codes, but also to polar codes. The advantage of P&S is that only one encoder/decoder pair is required for the entire SNR range, since the P&S positions are known in advance by the receiver, which effectively reduce the complexity of the CVQKD system.
4.4.2 Raptor codes
Raptor codes [88], a class of rateless codes with linear time encoding and decoding, have been used in the IR of CVQKD [92–95]. The factor graph of Raptor code is shown in Fig. 5(d).
Raptor codes need to cascade another coding process, such as LDPC codes, on top of the LT codes to achieve good coverage of the source sequence. This so called precoding can effectively reduce the complexity of compiling codes and improve the success rate of decoding. However, Raptor codes require a higher decoding complexity and longer decoding latency than other types of ECCs.
Raptorlike codes [96] can be constructed if the structure of the PCM of Raptor codes is kept unchanged. The check matrix of Raptorlike codes is constructed by the common matrix construction method of LDPC codes. The Raptorlike codes have not only the code rate compatibility, but also better decoding performance than LDPC codes. Zhou et al. have applied Raptorlike LDPC codes to IR [97].
4.4.3 Spinal codes
Spinal codes [89] are also rateless codes with a simple coding structure, it can adapt to timevarying channels without explicit bit rate selection. It have an efficient polynomialtime decoder, which achieves the Shannon capacity over both AWGN and BSC channels. In 2020, Wen et al. applied the spinal codes to IR and obtained good reconciliation performance [98].
4.5 Comparation of several ECCs
Table 4 shows the advantages and disadvantages of several ECCs compared with others.
5 Hardware acceleration
In a highspeed QKD system, the throughput of IR is one of the key factors governing the system key rate, as shown in Eq. (3). With the rapid development of experimental technology, the repetition rate of CVQKD systems has grown from MHz to GHz [39]. Correspondingly, highspeed and realtime IR is required to match the high repetition rate. In this section, we will introduce how to use hardwares to accelerate the reconciliation algorithm and improve the throughput.
By increasing the code length, the decoding performance of ECCs can be improved effectively. However, increase of the code length will also increase the computational complexity. The code length in classical communication and DVQKD is generally about 1k ∼ 10k [70, 99], while CVQKD often requires 100k ∼ 1M. The throughput of encoding/decoding algorithms based on general central processing units (CPUs) for long code is very limited.
In order to solve this problem, the powerful parallel computing capability of hardware can be employed to improve the computing speed. Since, the structure of PCMs of the ECCs is readily parallelized [100]. This is an important factor for achieving high throughput, and also makes encoders and decoders suitable for implementation on FPGA and GPU to achieve very high throughput [101].
FPGAs can be used for both the algorithm acceleration and control tasks. It is ideal and attractive for designing prototypes and the manufacturing of smallproductionrun devices. FPGAs based platforms have been demonstrated to facilitate quantum information processing, DVQKD prototypes [101–105], quantum algorithm [106, 107], and postprocessing DVQKD [108, 109]. The low power consumption makes FPGAs attractive for good integration ability. GPU provides floatingpoint computational precision with short development cycles and highbandwidth onchip memory. Table 5 compares the characteristics of FPGA and GPU.
5.1 FPGAbased acceleration
In 2020, Yang et al. achieved the highspeed hardwareaccelerated IR procedure on a FPGA chip by taking advantage of its superior parallel processing ability [55]. As shown in Fig. 6(a) and (b). Two different structures including multiplexing and nonmultiplexing are designed to achieve the tradeoff between the speed and area of FPGAs, so that an optimal scheme can be adopted according to the requirement of a practical system. Many FPGAbased LDPC decoders have been developed over past few decades [70]. However, these decoders do not meet the decoding performance requirements of IR in CVQKD systems. In 2021, Yang et al. proposed a highspeed layered SPA decoders with good performance and lowcomplexity for ultralong quasicyclic LDPC codes [72]. To reduce the implementation complexity and hardware resource consumption, the messages in the iteration process are uniformly quantified and the function \(\Psi (x)\) is approximated with secondorder functions. The decoder architecture improves the decoding throughput by using partial parallel and pipeline structures. A modified construction method of PCMs was applied to prevent read & write conflicts and achieve highspeed pipeline structure. The throughput of the LDPC decoder can be estimated by
where f is the clock frequency of FPGAs, q is the quasicyclic parameter, \({{N}_{node}}\) is the average number of nodes in each row of a basic matrix, and \({{N}_{iter}}\) is the average number of iterations to execute a decoding algorithm.
Recently, Lu et al. [110] designed an FPGAbased architecture for multidimensional reconciliation receiver module that can achieve high throughput according to system requirements with the toplevel logic diagram shown in Fig. 6(c). To implement the sender of multidimensional reconciliation, FPGAbased METLDPC decoders still needs to be designed and implemented. To reduce the complexity of hardware implementation, we can simplify the matrix operation based on the characteristic of fewer nonzero elements in the matrix family.
In principle, the throughput based on FPGAs can be improved by instantiating multiple modules on a chip, and the throughput can be further improved by increasing the parallelism of iterative decoding, etc. Currently, integrated photonic technology is develops rapidly [111, 112], and some research teams have tried to implement the CVQKD systems using silicon photonic chips [41, 43, 44]. To realize an overall integration, the IR module must also be able to be integrated and miniaturized. The advantages of easy integration and low power consumption of FPGAs make it a very competitive candidate.
5.2 GPUbased acceleration
GPUs are widely used in IR hardware acceleration for CVQKD due to their easy programming and short development cycle [49, 113–118]. The main difficulty of GPUs also lies in the implementation of storage and decoding algorithms. A GPU has many threads. If only one codeword is decoded at a time, the performance of the GPU cannot be fully utilized. Compute Unified Device Architecture (CUDA) can be utilized to implement multiple codewords parallel decoding algorithms, such as 64 codewords [115], 128 codewords [116], and 512 codewords [117]. Theoretically, the more parallel codewords, the greater throughput. However, the finite memory resources will limit the number of parallel codewords. The memory of GPUs is composed of global memory, constant memory, local memory, and so on. Due to the very large size of PCMs of IR, the developers need to optimize the memory structure of PCMs. To reduce latency, the iterative decoding messages are usually stored in global memory for coalesced access.
When selecting GPUs and according to the characteristics of IR, the main considerations are memory, threads, and clock frequency. Several models of GPUs have been employed in IR, such as NVIDIA GeForce GTX 1080/1650, TITAN Xp, and Tesla K40C accelerator card. With the advancement of technology, more and more onchip resources will be available, which will further increase the parallelism of the IR algorithm. To avoid some operations that are not easy to perform on GPU, a hybrid CPUGPU platform can be used [118].
6 Progress of IR
As shown in Fig. 7, a practical IR unit often contain three steps: designing a reconciliation scheme, selecting and optimizing the ECCs, and performing hardware acceleration. The early research on IR focused on improving the reconciliation efficiency to obtain high SKR per pulse, and after years of efforts, the reconciliation efficiency has now been able to reach more than 95%, which can meet most of the requirements of CVQKD systems. In recent years, the research of IR has gradually focused on highspeed and practicality with the rapid increase of the system repetition rate. In this section, we summarize the research progress of IR including: slice reconciliation, multidimensional reconciliation, and rateadaptive reconciliation.
6.1 Slice reconciliation
In Table 6, we summarize the current research advances of slice reconciliation. The first reconciliation algorithm used for CVQKD employs Turbo code [51] and its efficiency is less than 80%, which limits the maximum transmission distance to less than 20 km. By improving the LDPC based ECCs and optimizing the quantization of the Gaussian variable, the efficiency of the slice reconciliation gradually grows from less than 80% to above 95% [56, 119]. Recently, polar codes are employed in slice reconciliation with reconciliation efficiency of around 95% [83], notice that longer block length compared with that of LDPC are required. Yang et al. perform hardware acceleration of the slice reconciliation using FPGAs [55], the maximum throughput is higher than 100.9 M Symbols/s. However, the FER is relatively high (above 10%), and the rateadaptive slice reconciliation has not been implemented. Due to the rapid growing of the system clock rate (above GHz), the throughput needs to be further improved.
6.2 Multidimensional reconciliation
Multidimensional reconciliation has gained much attentions because it can support CVQKD systems over longer transmission distances. Table 7 shows a summary of multidimensional reconciliation research that have been reported. The reconciliation efficiency and throughput are gradually improving. Among them, the highest reconciliation efficiency can already reach 99%, at the cost of high FER. To reduce the FER, Feng et al. proposed a tBP decoding algorithm [62]. Their simulation results show that FER with the new decoding algorithm is superior to that with the conventional BP algorithm. To improve the throughput, GPUs are employed for hardware acceleration and have made great progress [49, 114–116]. For example, the throughput is increased from 7.1 Mb/s [49] to 64.11 Mb/s [116] when the SNR is 0.161. In addition, Lu et al. has implemented the encoding module of multidimensional reconciliation based on FPGAs [110].
6.3 Rateadaptive reconciliation
For practical QKD systems, the SNRs are not fixed and will inevitably fluctuate due to the transmission fluctuations of the quantum channel and variations of the QKD system itself. This requires that the optimal code rate of the ECCs used in the IR process should also be adjusted accordingly to guarantee the performance and security of the system. The traditional LDPC codes and polar codes are sensitive to the number of error bits in the bit string, which is unsuitable for the fluctuating channel in practical applications. It is impossible to solve the problem by storing a large number of check matrices with different bit rates. To cope with the timevarying quantum channel, it is necessary to adjust the code rate of ECCs in realtime by rateadaptive techniques (P&S) or rateless codes (Raptor codes and spinal codes).
Table 8 shows the reported rateadaptive IR. Most of the current works use the multidimensional reconciliation scheme. Wang et al. [74] and Jeong et al. [75] adopted METLDPC codes, which can achieve high reconciliation efficiency, but the SNR range covered by a single PCM with the fixed code rate is very small. Zhang et al. [82] and Cao et al. [85] adopted polar codes to realize rateadaptive reconciliation. Zhang et al. adopted the incremental freezing scheme of information bits in polar codes to fix the step size of frozen information bits, thus realizing the fixedstep coderate adjustment. In Ref. [85], the channel state is estimated after IR to estimate the SNR and calculate the code rate. Then, the positions of the punctured and shortened bits are determined. Notice that, the schemes of Wen et al. [98] and Fan et al. [124] can be applied to larger SNRs.
7 Applications of IR in CVQKD systems
At present, a number of teams have already applied IR to CVQKD systems. Lodewyck et al. [113] implemented a coherentstate CVQKD system with SKR of more than 2 kb/s on 25 km single mode fiber. In their QKD system, they used the slice reconciliation based on the LDPC codes and GPU. In 2009, they designed and realized a CVQKD prototype [125] that employs sophisticated ECCs for reconciliation. Thereafter, they extended the transmission distance to 80 km by using multidimensional reconciliation [126]. The multidimensional reconciliation is performed using METLDPC codes and achieved speeds up to several Mbps using an OpenCL implementation of BP decoding algorithm with flooding schedule on a GPU.
In 2015, Wang et al. [127] demonstrated CVQKD over 50 km fiber. To generate the secret keys at low SNR regime, they adopted multidimensional reconciliation and the rateadaptive LDPC code to perform key extraction offline. In 2021, they experimentally realized a passivestatepreparation CVQKD scheme and a slice reconciliation based on slicetype polar codes is employed [34].
In 2019, Zhang et al. [128] combine multidimensional reconciliation and METLDPC codes to achieve high reconciliation efficiency at low SNRs. They implement multiple code words decoding simultaneously based on GPU and obtain throughput up to 30.39 Mbps on a GPU. In Ref. [33], the authors use slice reconciliation with polar codes at 27.27 km and 49.30 km, multidimensional reconciliation with METLDPC codes at 69.53 km, 99.31 km, and 140.52 km, and multidimensional reconciliation with Raptor codes at the longest distance of 202.81 km, respectively.
Zhang et al. [41] demonstrate an integrated CVQKD system over a 2 m fiber link and generate secret keys with the slice reconciliation and LDPC codes. Furthermore, to prove the capability for longdistance CVQKD, they developed a rateadaptive reconciliation protocol based on multidimensional reconciliation and METLDPC codes.
In 2022, Wang et al. [76] designed three PCMs with code rates of 0.07, 0.06, and 0.03, which are suitable for transmission distances of 5, 10, 25 km in their QKD system, respectively. Jain et al. [35] obtained a reconciliation efficiency of \(\beta =94.3\%\) and \(FER = 12.1\)% for their experimental data (20 km long quantum channel) based on a multidimensional reconciliation scheme and METLDPC codes.
IR has applications not only in pointtopoint CVQKD systems, but also in CVQKD network systems [129, 130].
8 Challenges
Although great progress has been made for IR, there are still several challenges for future work.
Performance improvement. On the one hand, we should continue to study more advanced ECCs for IR to improve the reconciliation efficiency and reduce the FER further. Furthermore, the hybrid platform such as FPGAGPU should be investigated to give full play to the advantages of each platform. In this case, we can realize a better hardware acceleration performance and practicability (easy to use, low power consumption, and short development cycle) that are not attainable by using only a single platform. On the other hand, we know that several parameters of IR affect the performance of CVQKD systems from previous sections. Currently, most of the existing research works focus on improving one or two parameters. To design an efficient and practical IR unit, it is crucial to consider (improve and optimize) the reconciliation scheme, ECCs, hardware acceleration, and other aspects from a global perspective. In addition, new methods, such as artificial intelligence can be introduced to IR to reduce the decoding complexity [118].
Rateadaptive reconciliation. Most of the current research on rateadaptive reconciliation concentrate on multidimensional reconciliation. The rateadaptive slice reconciliation, which is suitable for shortrange key distribution needs to be investigated. High throughput rateadaptive reconciliation is critical for a high speed QKD system. To this end, the hardware structures need to be carefully designed according to the characteristics of the ratecompatible codes used. The main difficulties faced is how to improve the compatibility of codes in the hardware platform to make it compatible with different check matrices.
Standardization. With the rapid development of quantum information technology, its largescale applications gradually become possible. The relevant scientific and technological progress has make the standards formulations available [131]. As a key part of QKD technique, the corresponding standards of IR is required. The existing error correction standards for classical communications are not well suited for CVQKD. For example, ATSC 3.0 standard [123], an international broadcasting standard formulated by Advanced television systems committee (ATSC) in 2013, gives an encoding/decoding algorithm for LDPC codes. To establish the error correction standards for the CVQKD, one should identify the types of codes and the encoding/decoding algorithms that are most suitable for the IR of CVQKD.
9 Conclusions
In this paper, we have reviewed the IR in CVQKD. The rapid development of CVQKD technology has put forward growing requirements on IR. For future work, it is critical to further improve the overall performance of IR including throughput, efficiency, and FER. In this direction, various highperformance ECCs should be explored and developed to find the best candidate of IR. Furthermore, it is also necessary to efficiently adapt to timevarying channels for the IR. In terms of integration and miniaturization, special hardware architectures should be developed to dramatically accelerate the algorithm and reduce the power consumption. By taking the above measures, high performance and practical IR modules can be developed, which will pay the ways for large scale applications of CVQKD technology in future.
Availability of data and materials
Not applicable.
Abbreviations
 ATSC:

Advanced television systems committee
 AWGN:

Additive white Gaussian noise
 BP:

Belief propagation
 BSC:

Binary symmetric channel
 CVQKD:

Continuous variable quantum key distribution
 DR:

Direct reconciliation
 DVQKD:

Discrete variable quantum key distribution
 ECC:

Error correction code
 FER:

Frame error rate
 FPGA:

Fieldprogrammable gate array
 GPU:

Graphics processing unit
 IR:

Information reconciliation
 LBP:

Layered belief propagation
 LDPC:

Lowdensity paritycheck
 LLR:

Logarithmiclikelihood ratio
 MLC:

Multilevel coding
 MSA:

Minsum algorithm
 MSD:

Multistage decoding
 PA:

Privacy amplification
 PCM:

Parity check matrix
 PEG:

Progressive edge growth
 QKD:

Quantum key distribution
 RR:

Reverse reconciliation
 SKR:

Secret key rate
 SNR:

Signaltonoise ratio
 TRNG:

True random number generator
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Funding
Innovation Program for Quantum Science and Technology (No. 2021ZD0300703), National Natural Science Foundation of China (NSFC) (No. 62175138, 62205188), Shanxi 1331KSC, Fundamental Research Program of Shanxi Province (No. 202203021222232, 202203021211260), Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi Province (No. 2021L258) and Key Research and Development Program of Guangdong Province (No. 2020B0303040002).
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Yang, S., Yan, Z., Yang, H. et al. Information reconciliation of continuousvariables quantum key distribution: principles, implementations and applications. EPJ Quantum Technol. 10, 40 (2023). https://doi.org/10.1140/epjqt/s40507023001978
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DOI: https://doi.org/10.1140/epjqt/s40507023001978