Abstract
In this paper, we propose a design rule of ratecompatible punctured multiedge type lowdensity paritycheck (METLDPC) code ensembles with degreeone variable nodes for the information reconciliation (IR) of continuousvariable quantum key distribution (CVQKD) systems. In addition to the rate compatibility, the design rule effectively resolves the high errorfloor issue which has been known as a technical challenge of METLDPC codes at low rates. Thus, the proposed design rule allows one to implement ratecompatible METLDPC codes with good performances both in the threshold and lowerrorrate regions. The rate compatibility and the improved errorrate performances significantly enhance the efficiency of IR for CVQKD systems. The performance improvements are confirmed by comparing complexities and secret key rates of IR schemes with METLDPC codes whose ensembles are optimized with the proposed and existing design rules. In particular, the SNR range of positive secrecy rate increases by 1.44 times, and the maximum secret key rate improves by 2.10 times as compared to the existing design rules. The comparisons clearly show that an IR scheme can achieve drastic performance improvements in terms of both the complexity and secret key rate by employing ratecompatible METLDPC codes constructed with code ensembles optimized with the proposed design rule.
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Introduction
Quantum key distribution (QKD) systems allow two remote parties to share secret keys by utilizing quantum mechanics^{1}, which is known to provide unconditional security^{2,3}. In QKD systems, the secret keys are established by performing the following two phases: (1) exchanging quantum states through a quantum channel and (2) postprocessing through an authenticated classical channel^{4}. QKD systems are usually categorized into discretevariable QKD (DVQKD)^{1,2,4}, and continuousvariable QKD (CVQKD) systems^{5,6,7}, according to their modulation techniques adopted in the quantum state exchanges. In the DVQKD systems, the polarization of the singlephoton is modulated by the information while both the amplitude and phase quadrature of coherent state are modulated in the CVQKD systems. Recently, there have been extensive studies on practical CVQKD system developments^{8,9,10} since CVQKD systems can be readily deployed in the existing optical communication infrastructure^{11} and also overcome limitations of applicability of DVQKD systems, e.g., the requirement of a sophisticated singlephoton detector^{12}.
To achieve a higher key rate and longer operation range, it is important to increase the efficiency of information reconciliation (IR). In particular, CVQKD systems operate in the very low signaltonoise ratio (SNR) region where errorcorrecting codes (ECCs) for the IR must be designed at extremely low rates. It is technically challenging to design strong ECCs tailored for such a low SNR region, and there have been extensive efforts^{13,14,15} to improve the efficiency of IR by designing stronger ECCs of low rates. For instance, Raptor codes are designed at low rates for the IR in refs. ^{14,15,16,17} where Raptor codes have capacityapproaching performances. In addition, the rateless feature of Raptor codes enables the IR to maintain high efficiencies across a range of SNRs. However, Raptor codes require a high decoding complexity as compared to other types of ECCs, e.g., multiedgetype lowdensity paritycheck (METLDPC) codes, due to high check node degrees, which leads to excessively long decoding latency.
Meanwhile, METLDPC codes at low rates, e.g., 1/50, are employed in the IR for CVQKD systems due to their good errorcorrecting performances and more amenable decoding complexity^{18,19,20}. In ref. ^{18}, the authors demonstrated highspeed error correction for METLDPC codes utilizing a graphic processing unit (GPU). In ref. ^{19}, it was shown that a quasicyclic code construction of METLDPC codes is suitable for hardwareaccelerated decoding. In the studies^{18,19,20}, METLDPC codes are implemented based on degree distributions simply taken from an open literature^{13}, and the authors focus on the demonstrations of practical decoder implementation for METLDPC codes. In this work, we are instead interested in designing strong METLDPC codes.
There have been studies on METLDPC code design^{13,21,22} which pay their attention only to the optimization of the threshold performance. Thus, the designed METLDPC codes suffer from high error floors which limit the efficiency of the IR^{18,19}. In particular, the error floors are mainly due to an anomaly called the decoder errors, i.e., decoding into wrong codewords, which requires QKD systems to employ additional errordetection codes such as cyclic redundancy check (CRC) codes to confirm whether the decoded codeword is the transmitted one. In the case that a decodererror event happens, the QKD system may discard the shared randomness obtained via the first phase, i.e., exchanging quantum state over the quantum channel. It is also possible that the QKD system performs additional communications through a classical public channel to resolve the problem, which however leads to extra information leakage and eventually degrades the secret key rate in the key distillation. In addition, the CRC codes increase the hardware complexity. Thus, it is highly desirable to design METLDPC codes without suffering from the errorrate performance loss due to the high error floors caused by the decoder errors.
It was shown^{23} that smallweight codewords of METLDPC codes mainly induce the decoder errors, and thus the design of METLDPC code ensembles must be carried out to avoid smallweight codewords. Then, METLDPC codes without error floors can be implemented with ensembles that have diminishing average numbers of smallweight codewords with the growing code length. It is known that if an METLDPC code ensemble satisfy a certain condition so called, the tvalue condition, it has exponentially few codewords of small weight^{24}. Recently, the tvalue condition is further extended^{23} to METLDPC code ensembles with degreeone variable nodes which are essential for code ensembles of low rates to have good threshold performances^{25}.
The efficiency of IR depends on the code rate of ECC, namely the amount parity bits, which is determined by the quality of the quantum channel. It is often observed that the quality of the quantum channel varies in time due to various factors such as the number of photons affected by the attacker, noise variations, the thermally induced length fluctuations, the timing jitter, etc. Thus, to maximize the IR efficiency, the code rate must be adapted to the variation of quantum channel quality, which can be realized with multiple encoder and decoder pairs of different rates. While the scheme with multiple encoder/decoder pairs seems conceptually straightforward, it is not a pragmatic solution due to the growing complexity as the number of pairs increases. In addition, when the decoding is not successful, the transmitted codeword must be discarded, and a new codeword of a lower rate will be requested, which leads to a loss of IR efficiency.
In this work, we instead consider an IR scheme with a rateadaptive punctured ECCs which are derived by puncturing the parity bits of a code, called mother code. The puncturing provides a sequence of codes whose rates increase from that of the mother code depending on the number of punctured parity bits. In addition, the puncturing is carried out in the ratecompatible fashion^{26} where a code of higher rate is embedded in codes of lower rates. That is, punctured bits in a code of a rate must also be punctured in codes of higher rates than the rate. The beauty of ratecompatible punctured ECCs is that only one encoder/decoder pair is needed for the entire range of rates given the puncturing locations are a priori known to the receiver, which conveniently resolves the complexity issue in the scheme using multiple encoder/decoder pairs. More importantly, when the decoding fails, the scheme with ratecompatible punctured ECCs simply transmits some punctured parity bits which will be combined with the already received codeword resulting in a new codeword of a lower rate. The transmission of punctured parity bits can be repeated until the receiver successfully decodes the transmitted codeword. Thus, the scheme with ratecompatible punctured ECCs is an efficient solution to maximize the IR efficiency at reduced complexity. Recently, an IR scheme with punctured METLDPC codes was studied in ref. ^{20} where METLDPC codes are randomly punctured to adapt their rates. However, the recent work^{20} takes degree distributions of METLDPC codes from ref. ^{13} which will be shown to have poor errorfloor performances. In addition, the random puncturing in ref. ^{20} results in such poor errorfloor performances at all the rates derived from the mother code.
In this work, by utilizing the recent result^{23} for designing METLDPC codes with both good threshold and errorfloor performances, this work proposes an IR scheme using optimized ratecompatible punctured METLDPC codes. Since we consider the ratecompatible puncturing to realize the rate adaptability in this work, we simply call rateadaptive punctured METLDPC codes as ratecompatible METLDPC codes. The improvements of METLDPC codes in the threshold and errorfloor regions allow the IR scheme to achieve higher key rates and/or longer distances of CVQKD systems. The improved errorfloor performances enable one to design CVQKD systems without resorting to certain errordetection codes, which not only improves the efficiency of CVQKD systems but also reduces the complexity of an IR scheme. In particular, we propose a design of METLDPC code ensembles with degreeone variable nodes which have exponentially few smallweight codewords. The designed ensembles allow one to implement METLDPC codes with better performances in both the threshold and errorfloor regions at a reduced decoding complexity as compared to the ensembles based on the existing design rules^{18,19,25}. In addition, we will show how to design ratecompatible METLDPC codes while holding the tvalue condition over a range of code rates that the ratecompatible METLDPC codes support. The ratecompatible METLDPC codes can be utilized for implementing efficient IR schemes for CVQKD systems. The details of CVQKD system and the IR scheme considered in this work will be introduced in “Methods”. We will conduct performance comparisons among IR schemes with ratecompatible METLDPC codes and a fixedrate METLDPC code. For implementing the METLDPC codes, their code ensembles are optimized with the proposed design rule and existing design rules. The performance comparisons clearly show that significant performance improvements are achievable by employing the ratecompatible METLDPC codes using ensembles with the proposed design rule.
Results
The tvalue condition for METLDPC codes
In this work, we consider a Tanner graph of METLDPC code ensemble with degreeone variable nodes shown in Fig. 1 where the entire set of variable node classes, denoted by \({{{\mathcal{V}}}}\), is partitioned into the subsets, \({{{{\mathcal{V}}}}}_{1}\), \({{{{\mathcal{V}}}}}_{2}\), and \({{{{\mathcal{V}}}}}_{12}^{c}\), and \({{{{\mathcal{V}}}}}_{12}^{c}\) is the complement of the set \({{{{\mathcal{V}}}}}_{12}={{{{\mathcal{V}}}}}_{1}\cup {{{{\mathcal{V}}}}}_{2}\). Note that the variable node class \({{{{\mathcal{V}}}}}_{1}\) consists of two subclasses denoted by \({{{{\mathcal{V}}}}}_{1,p}\) and \({{{{\mathcal{V}}}}}_{1,np}\) which represent punctured and unpunctured variable nodes of degreeone, respectively. Similarly, the check node class \({{{{\mathcal{C}}}}}_{1}\) consists of two subclasses denoted by \({{{{\mathcal{C}}}}}_{1,p}\) and \({{{{\mathcal{C}}}}}_{1,np}\) which represent check nodes connected to the punctured and unpunctured variable nodes of degreeone, respectively. In this section, it is assumed that the all variable nodes of degreeone are unpunctured, and later in “Results”, we modify the result in ref. ^{23} to include the METLDPC codes with punctured variable nodes of degreeone.
In a similar manner, the entire sets of check node classes and edge types in Fig. 1, denoted by \({{{\mathcal{C}}}}\) and \({{{\mathcal{E}}}}\), respectively, are partitioned into \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{1}^{c}\), and \({{{{\mathcal{E}}}}}_{1}\), \({{{{\mathcal{E}}}}}_{2}\), and \({{{{\mathcal{E}}}}}_{12}^{c}\), respectively, where \({{{{\mathcal{E}}}}}_{12}^{c}\) is the complement of \({{{{\mathcal{E}}}}}_{12}={{{{\mathcal{E}}}}}_{1}\cup {{{{\mathcal{E}}}}}_{2}\), and \({{{{\mathcal{C}}}}}_{1}^{c}={{{\mathcal{C}}}}\setminus {{{{\mathcal{C}}}}}_{1}\). The blocks denoted by ET in Fig. 1 are uniform interleavers each of which permutes edges of a type. In particular, \({{{{\mathcal{V}}}}}_{1}\) is the set all variable node classes of degreeone, i.e., ∑_{j}d_{i,j} = 1 for \(i\in {{{{\mathcal{V}}}}}_{1}\), and \({{{{\mathcal{E}}}}}_{1}=\{j {d}_{i,j}=1,\ \forall \ i\in {{{{\mathcal{V}}}}}_{1}\}\). Note that \({{{{\mathcal{E}}}}}_{1}\) is the set of edge types corresponding to the variable nodes classes in \({{{{\mathcal{V}}}}}_{1}\). Meanwhile, \({{{{\mathcal{C}}}}}_{1}\) is the set of all check node classes which have check nodes incident with edges of types in \({{{{\mathcal{E}}}}}_{1}\), i.e., for every \(i\in {{{{\mathcal{C}}}}}_{1}\), \(\exists \ j\in {{{{\mathcal{E}}}}}_{1}\) such that g_{i,j} = 1. Note that the threshold of METLDPC code ensemble will not be defined if a check node is incident with more than one edge of a type in \({{{{\mathcal{E}}}}}_{1}\). Thus, each check node of a class in \({{{{\mathcal{C}}}}}_{1}\) has a single edge of a type in \({{{{\mathcal{E}}}}}_{1}\). Then, \({{{{\mathcal{E}}}}}_{2}\) is the set of all edge types for the edges incident to the check node of classes in \({{{{\mathcal{C}}}}}_{1}\) except for the ones of types in \({{{{\mathcal{E}}}}}_{1}\), i.e., for every \(j\in {{{{\mathcal{E}}}}}_{2}\), \(\exists \ i\in {{{{\mathcal{C}}}}}_{1}\) such that g_{i,j} > 0. The set \({{{{\mathcal{V}}}}}_{2}\) contains all variable node classes which have variable nodes incident with edges of types in \({{{{\mathcal{E}}}}}_{2}\), i.e., for every \(i\in {{{{\mathcal{V}}}}}_{2}\), \(\exists \ j\in {{{{\mathcal{E}}}}}_{2}\) such that d_{i,j} > 0. It is assumed that for each edge type of \(j\in {{{{\mathcal{E}}}}}_{12}^{c}\), there exists a check node of class \(i\in {{{{\mathcal{C}}}}}_{1}^{c}\) such that g_{i,j} ≥2, which is also considered in ref. ^{24}.
An METLDPC code ensemble can also be described with a pair of multinomials, and the pair of multinomials for the METLDPC code ensemble in Fig. 1 are given by
where the variable node classes in \({{{{\mathcal{V}}}}}_{1}\), \({{{{\mathcal{V}}}}}_{2}\), and \({{{{\mathcal{V}}}}}_{12}^{c}\) correspond to the first, second and third terms in ν(x), respectively, and the check nodes classes in \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{1}^{c}\) are represented by the first and second terms in μ(x), respectively. The code rate of METLDPC code ensemble^{27} is given by
For an METLDPC code ensemble, the average number of codewords of weight ℓ is expressed by the asymptotic exponential growth rate defined in Definition 1.
Definition 1 (the asymptotic exponential growth rate)
where A (ℓ) is the average number of codewords of weight ℓ, and w is the normalized weight.
Meanwhile, for smallweight codewords, i.e., w ≪ 1, Theorem 1 tells the asymptotic exponential growth rate, i.e., γ(w) in Definition 1.
Theorem 1 Ref. ^{22}. For t ≠ 0, we have
where \({\mathfrak{T}}\) is the set of all t values such that
where \(\tilde{u}=({\tilde{u}}_{1},{\tilde{u}}_{2},\ldots ,{\tilde{u}}_{ {{{{\mathcal{E}}}}}_{12}^{c} })\) (resp. \(\tilde{s}=({\tilde{s}}_{1},{\tilde{s}}_{2},\ldots ,{\tilde{s}}_{ {{{{\mathcal{E}}}}}_{12}^{c} })\)) is a vector whose elements, \({\tilde{u}}_{i}\)’s (resp. \({\tilde{s}}_{i}\)’s) are given by \({u}_{f({j}_{i})}\)’s (resp. \({s}_{f({j}_{i})}\)’s) for \(1\le i\le  {{{{\mathcal{E}}}}}_{12}^{c}\) and a bijective mapping \(f:{{{{\mathcal{E}}}}}_{12}^{c}\to \{1,2,\ldots , {{{{\mathcal{E}}}}}_{12}^{c} \}\), and \({{\Lambda }}^{\prime} (t)\) is a square matrix whose elements are given by
\(P^{\prime}\) is a square matrix whose elements are given by
\(\tilde{{{{\bf{0}}}}}=({o}_{1},{o}_{2},\ldots ,{o}_{{n}_{e}})\) is a vector of length n_{e}, \({o}_{m}=t\frac{{\sum }_{i\in {{{{\mathcal{C}}}}}_{1}}{\mu }_{i}{g}_{i,m}}{{\mu }_{{x}_{m}}({{{\bf{1}}}})}\) for \(m\in {{{{\mathcal{E}}}}}_{2}\) and zeros for the other elements.
It is shown in Theorem 1 that an METLDPC code ensemble with degreeone variable nodes has exponentially few codewords of small weights when the infimum of the solution set for the equation in Eq. (5) is larger than one, which is the tvalue condition and summarized in Definition 2.
Definition 2 (tvalue condition) For an METLDPC code ensemble, the infimum of \({\mathfrak{T}}\) is larger than unity. The infimum of \({\mathfrak{T}}\) will be called the tvalue of the ensemble.
Ratecompatible METLDPC codes for CVQKD systems
We will show that it is possible to design ratecompatible METLDPC codes with good errorrate performances in both the threshold and errorfloor regions, which is carried out by proving that there exists a sequence of punctured METLDPC code ensembles of rates with exponentially few codewords of small weights. To this end, we utilize the design rule in ref. ^{23} to optimize an METLDPC code ensemble with degreeone variable nodes for the threshold performance with the constraint of the tvalue condition. It was shown in ref. ^{23} that METLDPC codes based on the ensemble have good errorrate performances both in the threshold and errorfloor regions. The designed METLDPC code ensemble is called the mother code ensemble from which METLDPC code ensembles of higher rates are derived by puncturing parity bits of the mother code ensemble. Then, it will be shown that the punctured METLDPC code ensembles derived from the mother code ensemble also satisfy the tvalue condition regardless of the number of punctured parity bits if the mother code ensemble has a certain structure. In the next section, we will show that ratecompatible METLDPC codes with good errorfloor performances can be implemented using the punctured METLDPC code ensembles.
In the design of punctured METLDPC code ensembles, we puncture only degreeone variable nodes due to a few practical reasons. The Tanner graph in Fig. 1 shows that each degreeone variable node is incident to a different check node whose neighboring variable nodes are all unpunctured except for the degreeone variable node. Thus, the punctured degreeone variable nodes are onesteprecoverable (1SR)^{28}, i.e., recoverable in the first iteration of the beliefpropagation (BP) decoding. It was demonstrated in ref. ^{28} that punctured LDPC codes have good threshold performances when only 1SR variable nodes are punctured. In addition, the generation of coded bits for the punctured variable nodes can be performed with a linear complexity, which reduces the complexity of progressive parity bit generation and transmission.
For an METLDPC code ensemble with the degree distribution pair in Eq. (1), the degree distribution of the punctured METLDPC code ensemble becomes
where \({{{{\mathcal{V}}}}}_{1}={{{{\mathcal{V}}}}}_{1,p}\cup {{{{\mathcal{V}}}}}_{1,np}\), \({{{{\mathcal{V}}}}}_{1,p}\), and \({{{{\mathcal{V}}}}}_{1,np}\) indicate the sets of punctured and unpunctured degreeone variable node classes, respectively, and r_{0} and r_{1} represent the channels for the punctured the unpunctured variable nodes, respectively. The code rate of the punctured METLDPC code ensemble becomes
where π is the fraction of punctured degreeone variable nodes.
It should be noted that in the beliefpropagation (BP) decoding, the punctured degreeone variable nodes in \({{{{\mathcal{V}}}}}_{1,p}\) output their messages of zero loglikelihood ratio (LLR) value regardless of decoding iterations. In addition, for the check nodes incident to punctured variable nodes, the messages to unpunctured neighboring variable nodes are bounded by the LLR value of zero. Thus, the punctured degreeone variable nodes and their incident check nodes do not participate in the BP decoding, which allows us to exclude the punctured nodes from the ensemble. That is, the METLDPC code ensemble with punctured degreeone variable nodes can be expressed with an equivalent degree distribution pair that has only unpunctured variable nodes. Then, the equivalent METLDPC code ensemble is given by
where \({d}_{i,j}^{\prime}\) is the degree of the variable node class \(i\in {{{{\mathcal{V}}}}}_{2}\) for \(j\in {{{{\mathcal{E}}}}}_{2}\). Note that the removal of punctured variable nodes deletes some edges in \({{{{\mathcal{E}}}}}_{2}\) and check nodes of \({{{{\mathcal{C}}}}}_{1}\), which makes the degree \({d}_{i,j}^{\prime}\) less than or equal to d_{i,j}. i.e., \({d}_{i,j}^{\prime}\le {d}_{i,j}\). The degree \({d}_{i,j}^{\prime}\) is decided in such a way that the numbers of edges in \({{{{\mathcal{E}}}}}_{2}\) from variable nodes and check nodes are the same after removing the punctured variable nodes and their incident check nodes, which is so called the socket count equality^{23}.
For the equivalent code ensemble in Eq. (10), we have to test the tvalue condition^{23} to confirm that the punctured METLDPC code ensemble has exponentially few codewords of small weight. It is also especially important to know the maximum proportion of punctured bits below which the tvalue condition of the mother METLDPC code ensemble holds. In Theorem 2, we will prove that the tvalue condition of a mother code ensemble holds regardless of the proportion of punctured bits when the mother code ensemble has a certain structure. While the theorem is limited to METLDPC codes with the structure, it will be shown that some good mother METLDPC code ensembles can be readily designed even if the structural limit is imposed.
Theorem 2 For the METLDPC code ensemble of three edge types of \({{{{\mathcal{E}}}}}_{1}=\{1\}\), \({{{{\mathcal{E}}}}}_{2}=\{2\}\), and \({{{{\mathcal{E}}}}}_{12}^{c}=\{3\}\) with g_{i,2} = 0 for all \(i\in {{{{\mathcal{C}}}}}_{1}^{c}\), the following arguments are true:

1.
If a mother code ensemble satisfies the tvalue condition, punctured METLDPC code ensembles also satisfy the tvalue condition regardless of the amount of punctured bits.

2.
If a mother code ensemble does not satisfy the tvalue condition, none of punctured METLDPC code ensembles satisfies the tvalue condition.
Proof For an METLDPC code ensemble of three edge types of \({{{{\mathcal{E}}}}}_{1}=\{1\}\), \({{{{\mathcal{E}}}}}_{2}=\{2\}\), and \({{{{\mathcal{E}}}}}_{12}^{c}=\{3\}\) with g_{i,2} = 0 for all \(i\in {C}_{1}^{c}\), the ensemble in Eq. (1) can be rewritten as
For the ensemble in Eq. (11), the equality in Eq. (5) can be expressed as
Suppose that there exists a solution t ≤1 satisfying the equality in EQ. (12), which implies
where the lefthand side in Eq. (13) is obtained by replacing t in Eq. (12) with unity and is larger than or equal to the lefthand side in Eq. (12). Thus, when the inequality in Eq. (13) does not hold, the solutions of the equality in Eq. (12) must be larger than unity, i.e., \(\inf {\mathfrak{T}} \,>\, 1\). That is, the tvalue condition, i.e., \(\inf {\mathfrak{T}} \,>\, 1\), can be equivalently expressed as
If degreeone variable nodes in the ensemble of Eq. (11) are punctured with a fraction of π, the punctured ensemble can be represented as
For the degree distribution pair in Eq. (15), the equality in Eq. (5) can be expressed as exactly the same as the one in Eq. (12) except that d_{i,2} is changed to \({d}_{i,2}^{\prime}\), which however does not affect the inequality in Eq. (14). Thus, if the tvalue condition holds for the mother code, so does for all punctured code ensembles.◻
For arbitrary METLDPC code ensembles shown in Fig. 1, it is mathematically intractable to express in a closedform the maximum proportion of punctured parity bits below which the tvalue condition holds. However, it is sufficient to numerically test whether the punctured METLDPC code ensemble of the highest rate satisfy the tvalue condition since the ones of lower rates have additional parity bits, which does not induce smaller weight codewords.
Performance evaluations
In this section, we compare errorrate performances and efficiencies of IR schemes that have ratecompatible METLDPC codes and fixedrate METLDPC codes implemented using code ensembles with/without satisfying the tvalue condition. Refer to “Methods” for the CVQKD system in which the IR schemes are employed. We consider the multidimensional reconciliation with a dimension of 8. It is known in ref. ^{29} that the channel can safely be assumed to be a binaryinput additivewhiteGaussiannoise (BIAWGN) channel. For decoding METLDPC codes, we employ the sumproduct algorithm in which the maximum number of iteration is set to 1000. The iterative decoding terminates when all the parity checks are satisfied even before the iteration reaches the preset maximum number of iterations. The errorrate performances are measured in terms of both biterror rate (BER) and worderror rate (WER) which are evaluated at each SNR value by transmitting codewords until a hundred failed codewords are observed. In addition, the practicality of the METLDPC codes is compared in terms of three different metrics, i.e., the maximum variable node degree \({d}_{\max }\), the maximum check node degree \({g}_{\max }\), and a normalized edge density in ref. ^{30} which is defined as the average number of edges per message bit, i.e., \( {{{\mathcal{E}}}} /(R\cdot n)\) where \( {{{\mathcal{E}}}}\), R, and n are the total number of edges, code rate and code length, respectively. The maximum degrees are often used as a measure of decoding latency^{31} while the normalized edge density is adopted to measure the decoding complexity in ref. ^{30}.
First, we design a mother METLDPC code ensemble by optimizing the ensemble for the threshold performance^{22} with the constraint of the tvalue condition^{23}. The METLDPC code ensemble is optimized at a code rate of 0.02 for the BIAWGN channel, and is denoted by \({{{{\mathcal{C}}}}}_{1}\) in Table 1. For comparisons, we take a code ensemble in ref. ^{19}, and denote it by \({{{{\mathcal{C}}}}}_{2}\) in Table 1. Note that \({{{{\mathcal{C}}}}}_{1}\) has its tvalue of 1.0078 > 1 and thus satisfies the tvalue condition. Whereas \({{{{\mathcal{C}}}}}_{2}\) has its tvalue of 0.9743 < 1 and does not meet the tvalue condition. The tvalue condition tells when the number of smallweight codewords diminishes, which is obtained by a balance of degreeone and degreetwo of edge types at nodes. In the design of \({{{{\mathcal{C}}}}}_{2}\), the code optimization is carried out only for a good threshold performance, which more weighs the degreeone nodes and thus results in poor errorfloor performances. Based on the degree distributions of \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{2}\), two METLDPC codes of length 10^{6} are implemented with random paritycheck matrices, and they are denoted by \({{\mathbb{C}}}_{1}\) and \({{\mathbb{C}}}_{2}\), respectively. In this work, METLDPC codes are denoted by bold symbols, e.g., \({{\mathbb{C}}}_{1}\) and \({{\mathbb{C}}}_{2}\), while their ensembles are represented by script symbols, e.g., \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{2}\), respectively. Their BER and WER performances on the BIAWGN channel are compared in Fig. 2 where it is witnessed that \({{\mathbb{C}}}_{2}\) has a high errorfloor. The high errorfloor associated with \({{{{\mathcal{C}}}}}_{2}\) is mainly due to the decoder errors, i.e., decoding into wrong codewords, caused by smallweight codewords as predicted by the test of the tvalue condition. To substantiate the claim, we depict the decodererror rate (DER) in Fig. 2 where the DER and WER overlap each other in the errorfloor region. It should also be noted that the WER and BER of \({{\mathbb{C}}}_{2}\) have a wide gap, which is due to the fact that the decodererror events are caused by smallweight codewords^{23}. On the contrary, for the competing code, i.e., \({{\mathbb{C}}}_{1}\), we do not observe any decodererror event until its WER and BER reach 10^{−4} and 10^{−5}, respectively, and thus no errorfloor appears in Fig. 2.
Now, based on the two mother code ensembles, i.e., \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{2}\) in Table 1, we design ratecompatible METLDPC code ensembles which have their code rates between 0.02 and 0.025, equivalently, π ∈ [0, 0.2]. The equivalent degree distributions for the ratecompatible METLDPC codes at the highest code rate, i.e., 0.025, are described in Table 1, where the ones based on the code ensembles \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{2}\) are denoted by \({{{{\mathcal{C}}}}}_{1}^{\pi }\) and \({{{{\mathcal{C}}}}}_{2}^{\pi }\), respectively. Note that both the ensembles \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{2}\) are not designed for puncturing. It is possible to investigate into a design rule which also takes the puncturing into account, while it is beyond the scope of this work. Since the puncturing is carried out in the ratecompatible fashion^{26}, it enables one to progressively transmit additional parity bits when a decoding failure happens or an errordetection code finds out a decodererror event. Note that the ratecompatible METLDPC code using \({{{{\mathcal{C}}}}}_{1}\), i.e., \({{{{\mathcal{C}}}}}_{1}^{\pi }\) in Table 1, also satisfies the tvalue condition which is tested with the equivalent degree distribution pair in the \({{{{\mathcal{C}}}}}_{1}^{\pi }\) row of Table 1. It should be noted that the ensemble of the mother code, \({{{{\mathcal{C}}}}}_{1}\), has the structure discussed in Theorem 2, and thus the tvalue condition is always satisfied regardless of the amount of punctured parity bits. Meanwhile, it is shown in Table 1, the code ensemble using \({{{{\mathcal{C}}}}}_{2}\) does not satisfy the tvalue condition. It is also noticed in Table 1 that the thresholds of \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{1}^{\pi }\) are better than those of \({{{{\mathcal{C}}}}}_{2}\) and \({{{{\mathcal{C}}}}}_{2}^{\pi }\) while both \({{{{\mathcal{C}}}}}_{1}\) and \({{{{\mathcal{C}}}}}_{1}^{\pi }\) have lower complexities in terms of all the complexity measures, i.e., edge density, maximum variable node, and check node degrees. Thus, the ratecompatible METLDPC codes constructed with ensembles using the proposed design rule not only outperform the ratecompatible METLDPC codes using the existing design rule but also have practical advantages.
By puncturing the METLDPC codes, i.e., \({{\mathbb{C}}}_{1}\) and \({{\mathbb{C}}}_{2}\) in Fig. 2, two ratecompatible METLDPC codes of rate 0.025 (equivalently, π = 0.2) are implemented and evaluated in terms of BER and WER on the BIAWGN channel in Fig. 3 where the ratecompatible METLDPC codes are denoted by \({{\mathbb{C}}}_{1}^{\pi }\) and \({{\mathbb{C}}}_{2}^{\pi }\). Note that while the ratecompatible METLDPC codes \({{\mathbb{C}}}_{1}^{\pi }\) and \({{\mathbb{C}}}_{2}^{\pi }\) are obtained by puncturing their mother codes \({{\mathbb{C}}}_{1}\) and \({{\mathbb{C}}}_{2}\), the asymptotic behaviors of \({{\mathbb{C}}}_{1}^{\pi }\) and \({{\mathbb{C}}}_{2}^{\pi }\), e.g., thresholds and tvalue conditions, are given by the degree distributions in the \({{{{\mathcal{C}}}}}_{1}^{\pi }\) and \({{{{\mathcal{C}}}}}_{2}^{\pi }\) rows of Table 1, respectively. As predicted by Theorem 2, the ratecompatible METLDPC code, \({{\mathbb{C}}}_{2}^{\pi }\) suffers from the error floors as its mother code, i.e., \({{\mathbb{C}}}_{2}\) in Fig. 2, does. Whereas the errorfloor does not appear in the error rates of \({{\mathbb{C}}}_{1}^{\pi }\) as predicted by the test of the tvalue condition for the equivalent degree distribution pair. The comparison of errorrate performances in Fig. 3 confirms the results in “Results”.
The key rate of CVQKD systems is often^{9,19} assumed as
However, a recent work^{16} shows that the key rate formula in Eq. (16) does not reconcile with the results of quantum information theory in some situations. In particular, the issue can happen when the error rates of ECCs are relatively high, which is frequently encountered in longdistance CVQKD systems. Thus, instead of the key rate in Eq. (16), as suggested in ref. ^{16}, we use a bound on the key rate which is given by
where β is the IR efficiency defined as R_{π}/I_{AB}, R_{π} is the code rate, π is the maximum fraction of punctured bits when the IR succeeds, I_{AB} is the mutual information of the virtual Gaussian channel, and χ_{BE} is the Holevo bound on the information leaked to the eavesdropper, Eve^{11}. The secret key rate depends on various physical parameters such as the length and standard loss of fiber and homodyne detector efficiency, etc. In this work, we take the physical parameters from ref. ^{19} where the noise in the quantum channel denoted by χ_{tot} modeled as the sum of noises from the fiber and detector denoted by χ_{line} and \({\chi }_{\det }\), respectively. Then, the noise due to fiber of length ℓ with a transmittance T = 10^{αℓ/10} is given by χ_{line} = 1/T − 1 + ε where α = 0.2dB/Km is the standard loss of a singlemode fiber, and the excess channel noise (measured in shot noise units) is ε = 0.01 for 0 Km ≤ ℓ ≤100 Km, and ε = 0.01 + 0.001 × (ℓ − 100) for 100 Km ≤ ℓ ≤170 Km. Meanwhile, the noise in the homodyne detector is given by \({\chi }_{\det }=(1+{V}_{{{{\rm{el}}}}})/\eta 1\) where η and V_{el} represent the homodyne detector efficiency and additive electronic noise, respectively, and it is assumed that η = 0.606 and V_{el} = 0.041. Then, the total noise in the quantum channel follows as \({\chi }_{{{{\rm{tot}}}}}={\chi }_{{{{\rm{line}}}}}+{\chi }_{\det }/T\). In addition, the SNR of the virtual Gaussian channel is expressed as V_{A}/(1 + χ_{tot}) where V_{A} is a modulation variance of Alice and has to be optimized to achieve the highest key rate^{13}.
We define (1 − WER) × β in Eq. (17) as an effective efficiency which depends on both the WER and the efficiency of ECC, i.e., the maximum code rate at which the IR is successfully performed. We consider IR schemes with ratecompatible METLDPC codes by puncturing \({{\mathbb{C}}}_{1}\) and \({{\mathbb{C}}}_{2}\) in Fig. 2 as their mother codes. In addition, an IR scheme with the METLDPC code at a fixed rate of 0.02 denoted by \({{\mathbb{C}}}_{2}\) in Fig. 2. The effective efficiencies of three schemes are compared in Fig. 4 where RC \({{\mathbb{C}}}_{1}\) and RC \({{\mathbb{C}}}_{2}\) indicate the IR schemes with the ratecompatible METLDPC codes using \({{\mathbb{C}}}_{1}\) and \({{\mathbb{C}}}_{2}\), respectively. Meanwhile, the IR scheme with the fixedrate METLDPC code of \({{\mathbb{C}}}_{2}\) is denoted by \({{\mathbb{C}}}_{2}\) in Fig. 4. The performance comparison in Fig. 4 is carried out over a range of SNR values over which the maximum effective efficiencies of three schemes are observed. In practice, the SNR values are adjusted by controlling the modulation variance, V_{A} at a given length of fiber, ℓ. In Fig. 4, the efficiencies of the three schemes are depicted with the curves in blue, red, and black, respectively. In the IR scheme with \({{\mathbb{C}}}_{2}\), the shared randomness obtained via the quantum channel will be discarded when the decoding for ECC fails. Meanwhile, the IR schemes with RC \({{\mathbb{C}}}_{1}\) and RC \({{\mathbb{C}}}_{2}\) transmit additional parities when Alice requests on the decoding failure, which results in significant improvements of efficiency. However, the scheme with RC \({{\mathbb{C}}}_{2}\) suffers from decodererror events, which requires additional parity bits for the CRC code. In the case that the decoding for CRC code detects a decodererror event, i.e., \(u^{\prime} \ne u\), the scheme discards the shared randomness obtained via the communication over the quantum channel. On the contrary, the scheme with RC \({{\mathbb{C}}}_{1}\) has no decodererror event as the consequences of Theorem 2 promise. The comparisons in Fig. 4 quantitively demonstrate the performance improvements obtained by employing the ratecompatible METLDPC codes based on the ensemble satisfying the tvalue condition. That is, the scheme with RC \({{\mathbb{C}}}_{1}\) has a clear performance advantage over a wide range of SNR values as compared to the other two schemes, i.e., RC \({{\mathbb{C}}}_{2}\) and \({{\mathbb{C}}}_{2}\).
It should be mentioned that the comparison between the schemes with \({{\mathbb{C}}}_{2}\) and RC \({{\mathbb{C}}}_{2}\) has a crossover at the SNR of −15dB. In the region of SNR less than −15dB, the variation of efficiency, β is relatively small while the WER of \({{\mathbb{C}}}_{2}\) drastically improves with the growing SNR value since the threshold of error rate for \({{\mathbb{C}}}_{2}\) starts at around −15.3dB as shown in Fig. 2. Meanwhile, the improvement of WER for RC \({{\mathbb{C}}}_{2}\) is limited since the retransmissions of parity bits lead to decodererror events. This is why the scheme with \({{\mathbb{C}}}_{2}\) has better effective efficiency in the region of SNR less than −15dB. Note that efficiency is defined as the ratio of code rate and channel capacity. Thus, the efficiency β of the scheme with \({{\mathbb{C}}}_{2}\) sharply decreases as the SNR value further increases passing the crossover point, i.e., −15dB, considering that the capacity grows while the rate is fixed. On the contrary, it is not serious for the scheme with RC \({{\mathbb{C}}}_{2}\) due to the rate adaptability, i.e., the code rate adapts to the channel quality. It is noticed in Fig. 4 that the scheme with RC \({{\mathbb{C}}}_{2}\) outperforms the one with \({{\mathbb{C}}}_{2}\). As compared to the two schemes with \({{\mathbb{C}}}_{2}\) and RC \({{\mathbb{C}}}_{2}\), the scheme with \({{\mathbb{C}}}_{1}\) does not suffer from the decodererror events while taking the advantage of rate adaptability, which provides the performance superiority over the range of SNR values over the competing IR schemes, i.e., the ones with \({{\mathbb{C}}}_{2}\) and RC \({{\mathbb{C}}}_{2}\).
The key rate in Eq. (17) is evaluated in terms of bits per pulse for the three IR schemes in Fig. 5 where the length of fiber is assumed to be ℓ = 90 Km. It is clearly witnessed that the IR scheme with RC \({{\mathbb{C}}}_{1}\) outperforms the other two schemes over a wider range of SNR values. In addition, Fig. 6 shows that the IR scheme with RC \({{\mathbb{C}}}_{1}\) consistently performs better than the other schemes at different lengths of fiber of ℓ = 80 Km and 85 Km. In practice, the modulation variance V_{A} is determined to make the CVQKD system operate at a certain SNR value of the virtual Gaussian channel. The adjustment of V_{A} requires a precise channel estimation of the quantum channel, which is hard to achieve in practice. Thus, the SNR value has some variations, and the IR scheme must be designed robust to the SNR variation. In this sense, the IR scheme with RC \({{\mathbb{C}}}_{1}\) has clear advantages of both performance and practicality.
Discussion
In this paper, we proposed a design rule of multiedge type lowdensity paritycheck code ensembles with degreeone variable nodes. It was shown that the design rule allows one to implement ratecompatible METLDPC codes with good performances both in the threshold and lowerrorrate regions. It is also demonstrated that the ratecompatible METLDPC codes can improve the efficiency of information reconciliation for CVQKD systems.
Methods
Information reconciliation of QKD system
In this work, we consider the IR scheme for CVQKD systems introduced in ref. ^{15} where the scheme employs ratecompatible errorcorrecting codes, i.e., ratecompatible METLDPC codes in this work. The IR scheme is depicted in Fig. 7 where Alice transmits Gaussian random variables \({x}_{i} \sim {{{\mathcal{N}}}}(0,{\sigma }_{A}^{2})\) for i = 1, 2, …, d over the quantum channel, and for each Gaussian random variable x_{i}, Bob receives a noisy observation y_{i} = x_{i} + n_{i} from the quantum channel where \({n}_{i} \sim {{{\mathcal{N}}}}(0,{\sigma }_{n}^{2})\) and \({\sigma }_{n}^{2}\) indicates the noise power. Then, Alice and Bob have the correlated random vectors x = (x_{1}, x_{2}, …, x_{d}) and y = (y_{1}, y_{2}, …, y_{d}), respectively, which are normalized to \(x^{\prime} =x/  x \) and \(y^{\prime} =y/  y \) where ∣∣x∣∣ and ∣∣y∣∣ are the Euclidean norms of the vectors x and y, respectively, and d is called the dimension of multidimensional reconciliation^{29}.
In the reverse reconciliation^{32}, Bob generates a uniformly random binary sequence u from the quantum random number generator (QRNG), and encodes the sequence u into a codeword ξ, i.e., a codeword of METLDPC code in this work. Then, Bob divides ξ into subgroups of coded bits of length d. Suppose that one of the subgroups is denoted by c = (c_{1}, c_{2}, …c_{d}) which is converted to a spherical codes \(c^{\prime}\) as follows:
Note that \(c^{\prime}\) is uniformly distributed on the unit sphere in the d dimensional Euclidean vector space. For a pair of vectors, \(c^{\prime}\) and \(y^{\prime}\), Bob calculates the linear mapping \(M(y^{\prime} ,c^{\prime} )\) in ref. ^{29} such that
The mapping \(M(y^{\prime} ,c^{\prime} )\) is transmitted to Alice over the classical channel. When Alice receives the mapping \(M(y^{\prime} ,c^{\prime} )\), she performs \(M(y^{\prime} ,c^{\prime} )\cdot x^{\prime} =c^{\prime} +e\) where e follows a Gaussian distribution with zero mean^{29}. According to ref. ^{29}, as the dimension denoted by d grows, the d consecutive instances of the physical Gaussian channel, i.e., the quantum channel, are reformulated to d copies of a virtual BIAWGN channel^{29}. Since this work focuses on the benefit of ratecompatible METLDPC codes, it is assumed for simplicity that the dimension is fixed to d = 8 at which the quantum channel can be modeled as a BIAWGN channel as shown in Fig. 7. Since the gain in key rate is mainly due to the proposed coding scheme, the gain is also achievable with different dimensions. The transmission of subgroup c is repeated until the entire codeword ξ is transmitted. Then, in practical systems, the received signal \(c^{\prime} +e\) is often represented as a loglikelihood ratio (LLR) vector denoted by L in Fig. 7, and the LLR vector L is fed to the ECC decoder as its input.
In this work, a ratecompatible METLDPC code is assumed as the ECC in Fig. 7, and the rate of ECC is set to the maximum value or the capacity of the virtual BIAWGN channel by puncturing parity bits. If the decoding at the Alice side fails, she requests for additional parity bits. Upon the request, Bob transmits additional parity bits by sending a sequence of mappings corresponding to the punctured parity bits to be transmitted. The request and transmission will be continued until Alice successfully obtains her estimate of the message \(u^{\prime}\). In some cases, the estimate \(u^{\prime}\) is different from the true message u, which can be detected by employing an additional errordetection code such as a CRC code. In Fig. 7, the parity bits for CRC code is denoted by r. Note that the transmission of parity bits for the CRC code degrades performance of the IR scheme, which can be avoided by carefully designing the ECC. More details of the IR of CVQKD can be found in refs. ^{15,19}.
Data availability
The authors can confirm that the data supporting the findings of this study are available within the article.
Code availability
The code that supports the findings of this study is available on reasonable request from the author.
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Acknowledgements
This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) support program (IITP20212018001402) supervised by the IITP (Institute for Information & Communications Technology Planning & Evaluation) and was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) (No. 2018000831, A Study on Physical Layer Security for Heterogeneous Wireless Network). This was also supported by BK21 FOUR program.
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S.J. and H.J. developed the theoretical framework with which he designed ratecompatible METLDPC codes for CVQKD systems. Meanwhile, J.H. supervised this work and also contributed to writing the manuscript. All authors discussed the results and contributed to the writing of the manuscript.
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Jeong, S., Jung, H. & Ha, J. Ratecompatible multiedge type lowdensity paritycheck code ensembles for continuousvariable quantum key distribution systems. npj Quantum Inf 8, 6 (2022). https://doi.org/10.1038/s41534021005099
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DOI: https://doi.org/10.1038/s41534021005099
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