# Non-Gaussian reconciliation for continuous-variable quantum key distribution

Xiangyu Wang1<EMAIL_ADDRESS>Menghao Xu1 Yin Zhao1 Ziyang Chen2
<EMAIL_ADDRESS>Song Yu1 Hong Guo2 1State Key Laboratory of Information
Photonics and Optical Communications, Beijing University of Posts and
Telecommunications, Beijing 100876, China  2State Key Laboratory of Advanced
Optical Communication Systems and Networks, School of Electronics and Center
for Quantum Information Technology, Peking University, Beijing 100871, China

###### Abstract

Non-Gaussian modulation can improve the performance of continuous-variable
quantum key distribution (CV-QKD). For Gaussian modulated coherent state CV-
QKD, photon subtraction can realize non-Gaussian modulation, which can be
equivalently implemented by non-Gaussian postselection. However, non-Gaussian
reconciliation has not been deeply researched, which is one of the key
technologies in CV-QKD. In this paper, we propose a non-Gaussian
reconciliation method to obtain identical keys from non-Gaussian data.
Multidimensional reconciliation and multi-edge type low density parity check
codes (MET-LDPC) are used in non-Gaussian reconciliation scheme, where the
layered belief propagation decoding algorithm of MET-LDPC codes is used to
reduce the decoding complexity. Furthermore, we compare the error correction
performance of Gaussian data and non-Gaussian data. The results show that the
error correction performance of non-Gaussian data is better than Gaussian
data, where the frame error rate can be reduced by 50% for code rate 0.1 at
SNR of 0.1554 and the average iteration number can be reduced by 25%.

††preprint: APS/123-QED

##  I INTRODUCTION

Secure communication needs secret keys. However, the classical key generation
algorithm based on computational complexity is seriously threatened by quantum
computer and new mathematical algorithm. In order to solve this problem,
scholars have proposed quantum key distribution (QKD) protocols based on the
basic principles of quantum physics, which is one of the most mature quantum
information technology [1, 2, 3]. QKD allows two separate parties (Alice and
Bob) to establish unconditional secure keys through an unsecure quantum
channel which maybe controlled by potential eavesdroppers (Eve). According to
the encoding dimension of quantum states, QKD is divided into two branches,
discrete-variable (DV) QKD [4, 5] and continuous-variable (CV) QKD [6, 7, 8,
9]. These two kinds of protocols both have unconditional security [10, 11, 12,
13], in which Gaussian modulated coherent states CV-QKD protocols have the
advantages of being compatible with classical coherent optical communication
technology [14, 15]. However, due to the imperfection of the practical system
devices and the reconciliation efficiency of postprocessing, the transmission
distance of CV-QKD system is limited [16, 17, 18]. Therefore, improving the
secret key rate at given distance [19, 20] and extending the transmission
distance of the system are two improtant development trends in the field of
CV-QKD [9, 21, 22].

Recently, CV-QKD has made great progress in theory [23, 24, 25] and experiment
[26, 27]. The transmission distance has been significantly improved due to the
optimization of experiment setups and the improvement of reconciliation
efficiency [21, 28]. Some new protocols have been proposed to improve the
performance of CV-QKD system, such as the noiseless linear amplification [29]
and photon subtraction [30, 31]. These two quantum operation can extend the
transmission distance. However, due to the imperfection of actual devices and
other factors, it is difficult to implement in physical system. Thus, it is
hard to achieve the ideal effect. To solve these problems, some postselection
protocols are proposed. Through Gaussian postselection, virtual noiseless
linear amplification can be realized [32], it has been demonstrated
experimentally [33]. The physical realization of photon subtraction operation
requires an ideal single photon detector. Therefore, the implementation cost
will be increased, the effect will be reduced due to the imperfection of the
actual devices. Non-Gaussian postselection is proposed to implement virtual
photon subtraction, which avoids the use of single photon detector and complex
physical operation [34]. The raw data after Gaussian postselection is still
following Gaussian distribution, while for the virtual photon subtraction
inside Alice the raw data of Alice after non-Gaussian postselection is no
longer following Gaussian distribution.

Figure 1: The PM scheme of CV-QKD with non-Gaussian postselection in Alice.
$P(k|x_{A},p_{A})$ is the postselection filter function. $\gamma$ are Alice’s
raw data. $\left|\alpha\right\rangle$ is output quantum state. $T_{C}$ is the
transmittance of quantum channel. $\epsilon$ is excess noise. Hom: homodyne
detection. Het: heterodyne detection. The side information transmitted from
the classical authentication channel includes the selection results of Alice
in postselection process and the information used in postprocessing process.

Postselection algorithm can realize virtual physical operation, which greatly
reduces the implementation complexity. However, the corresponding classical
postprocessing part has not been deeply studied. There are still errors
between Alice’s and Bob’s raw data after postselection. Therefore, it is
necessary to correct the errors by using channel coding and decoding
technology to obtain consistent keys. The raw data are continuous variables in
CV-QKD. Therefore, it needs to be transformed into binary data that can be
encoded through mapping transformation firstly. Generally, there are mainly
two methods, slice reconciliation [35] and multidimensional reconciliation
[36]. The former is usually used in the case of high signal-to-noise ratio
(SNR), while the latter is used in the case of low SNR. The common error
correction codes used in CV-QKD are Raptor codes [37], Polar codes [38], and
low density parity check codes (LDPC) [39] etc. Multi-edge type (MET) LDPC
codes can achieve good error correction performance under extremely low SNRs.

In this paper, we mainly focus on the non-Gaussian reconciliation in CV-QKD.
The raw data after photon subtraction no longer follow Gaussian distribution,
the distribution varies with the number of photon subtraction. Firstly, we
give the method which realizes the transformation from Gaussian distribution
data to non Gaussian data through postselection filter function. This process
is very important, which affects the amount of raw data saved after non-
Gaussian postselection, it has an important impact on the secret key rate of
CV-QKD system. Multidimensional reconciliation and MET-LDPC codes are used for
the non-Gaussian reconciliation in reverse reconciliation system. We introduce
layered belief propagation decoding algorithm [40] into non-Gaussian data
error correction, which greatly reduces the complexity of the postprocessing
decoding process and does not increase the frame error rate (FER) of decoding.
Furthermore, we test the error correction performance of the non-Gaussian
reconciliation. Although the noise and Bob’s raw data still obey Gaussian
distribution, Alice’s data converge to medium amplitude after non-Gaussian
postselection, so the anti-noise performance of the system is improved. It can
be seen from the results that the FER of non-Gaussian data error correction is
obviously lower than that of Gaussian data under the same conditions, the
average number of iterations is also significantly reduced by using layered
decoding algorithm.

The rest of the paper is organized as follows. In Sec. II, we introduce some
basics of the non-Gaussian postselection, information reconciliation of CV-
QKD, presenting a postselection method to convert Gaussian distribution data
to non-Gaussian distribution data, proposing the non-Gaussian reconciliation
algorithm based on multidimensional reconciliation and MET-LDPC codes. In Sec.
III, we present the data distribution under different virtual photon
subtraction numbers, the performance tests of information reconciliation on
the non-Gaussian data after postselection under different virtual photon
subtraction numbers, and the error correction performance comparation with
Gaussian data. In Sec. IV, we draw the conclusion of this paper.

##  II Non-Gaussian reconciliation in CV-QKD

Non-Gaussian operation can increase the entanglement of the Gaussian entangled
states. As a non-Gaussian operation, it has been proposed that photon
subtraction in CV-QKD can increase the transmission distance. It has been
proved that the entanglement-based scheme and the corresponding prepare-and-
measure scheme is secure [31]. Simultaneously, the feasibility and security of
virtual photon subtraction scheme through non-Gaussian postselection have also
been proved [34]. In this section, we first introduce the basic of the non-
Gaussian postselection, then propose the postselection method to convert
Gaussian distribution data to non-Gaussian distribution data, finally present
the non-Gaussian reconciliation scheme based on multidimensional
reconciliation and MET-LDPC codes.

###  II.1 Non-Gaussian postselection in CV-QKD

The security of entanglement-based model CV-QKD photon subtraction protocol
has been proved, the security of the corresponding prepare-and-measure (PM)
model with equivalent postselection as virtual photon subtraction has also
been proved [34]. Different from Gaussian quantum state protocol, non-Gaussian
quantum state protocol can not have a symplectic covariance matrix, so we
can’t use von-Neumann entropy to derive the Holevo boundary for secret key
rate directly. However, we can estimate the secret key rate through the
physical model equivalent to virtual photon subtraction. This equivalent
physical model has been completed in our previous work [34]. The main idea is
based on the optimality of Gaussian attacks [10, 41, 42]. The eavesdropper
will not get more information from the non-Gaussian quantum state than the
Gaussian quantum state, so the secret key rate of the non-Gaussian protocol is
less than that of the corresponding Gaussian protocol. To ensure unconditional
security, we can use the secret key rate of the corresponding Gaussian
protocol as the lower bound of the secret key rate of the non-Gaussian
protocol. In addition, the probability of photon subtraction success should
also be considered when estimating the secret key rate of non-Gaussian
protocol.

Most of the implementation of CV-QKD is based on PM scheme, which does not
need to prepare entangled states, so it is easy to implement in experiment.
The PM scheme of CV-QKD with non-Gaussian postselection in Alice is shown in
Fig. 1.

In the PM scheme, Alice generates coherent states $\left|\alpha\right\rangle$,
where $\alpha=\sqrt{2T\lambda^{2}}\gamma$, $\gamma=x_{A}+ip_{A}$. $x_{A}$ and
$p_{A}$ are both randomly selected from a Gaussian distribution data set with
mean 0 and variance $V_{A}$. $T$ is related to the transmittance of photon
subtraction, $\lambda^{2}=\dfrac{V-1}{V+1}$, and $V$ is variance of the two-
mode squeezed vacuum state. Then after the coherent states are prepared, they
are sent to Bob through quantum channels. Bob performs homodyne or heterodyne
detection according to the type of protocol after receiving the quantum
states, the measurement results are recorded as $x_{B}$ and $p_{B}$. If Bob
performs homodyne detection, he informs Alice the quadratures ($x$ or $p$) he
measures. Alice keeps the same quadratures with Bob. While in the case of
heterodyne detection, both quadratures of $x$ and $p$ are kept.

After the quantum measurement and base sifting step, Alice chooses to accept
part of the raw data according to the non-Gaussian postselection filter
function with probability $P(k|x_{A},p_{A})$, the probability function of
subtracting $k$ photons is described by

$\begin{split}P(k|x_{A},p_{A})=\frac{1}{k!}[\dfrac{(1-T)\lambda^{2}}{2}(x_{A}^{2}+p_{A}^{2})]^{k}\times\\\
\exp[-\dfrac{(1-T)\lambda^{2}}{2}(x_{A}^{2}+p_{A}^{2})].\end{split}$ (1)

Then she reveals the selection results to Bob. Bob keeps the corresponding raw
data. After the non-Gaussian postselection, Alice’s raw data no longer follow
Gaussian distribution. But the distribution of Bob’s raw data remains
unchanged. The probability density function of the raw data before and after
non-Gaussian postselection is shown in Fig. 2. The black solid line represents
original Gaussian distribution function of Alice’s raw data. The red, green,
blue solid lines represent non-Gaussian distribution function of Alice’s data
after virtual photon substraction with $k=1,2,3$ respectively. As shown in
Fig. 2, for the raw data of Alice, the probability of some data in Gaussian
distribution is lower than that in non-Gaussian distribution. Therefore, it is
impossible to directly sample non-Gaussian distribution data from Gaussian
distribution data in this situation.

Figure 2: The probability density function of Gaussian distribution and Non-
Gaussian distribution after postselection. The X-axis is the value of raw
data, the Y-axis is the corresponding density. $g(x)$ is the probability
density function of Gaussian distribution. $c$ is a constant for effective
sampling. $f_{1}(x)$, $f_{2}(x)$, $f_{3}(x)$ are the probability density
function of non-Gaussian distribution after virtual photon subtraction with
$k=1,2,3$ separately. The modulation variance is set to 20.

To solve this problem, the acceptance-rejection sampling method is used to
convert the Gaussian distribution data to non-Gaussian distribution data,
which is a sampling method from probability density function. This method is
to extract subsequence of random variables from a sequence with a specific
distribution according to a rule and make them meet the given probability
distribution. Suppose the random variable is $x$, its value range is $[a,b]$,
its probability density function $f(x)$ of non-Gaussian distribution is
bounded, which satisfies max$\\{f(x)|a\leqslant x\leqslant b\\}=A.$ For
effective sampling, the probability density function of Gaussian distribution
is required to be greater than that of non-Gaussian distribution for any $x$.
However, as shown in Fig. 2, the probability density function of raw data
$g(x)$ does not satisfy the condition that it is always greater than $f(x)$
($f_{1}(x)$, $f_{2}(x),f_{3}(x)$) for any $x$. In order to satisfy the
condition, we construct a new probability density function which satisfies

$cg(x)\geqslant f_{k}(x),x\in[a,b],$ (2)

where $c$ is a constant. The purple solid line represents the probability
density function of $cg(x)$, it can be seen that the probability is greater
than that of $f(x)$ for any $x$. The value of $c$ should be as small as
possible to improve sampling efficiency which can be defined as

$E=\dfrac{1}{m}.$ (3)

It represents the average number of original distribution random variables $m$
required to obtain a random variable with a specific distribution. In our
case, it is equivalent to the success probability of the overall non-Gaussian
postselection.

For a raw data $x$ of Alice, she randomly generates a random number $d$ that
obeys a uniform distribution in the interval of $[0,cg(x)]$, where
$d=cg(x)\xi,\xi\in U[0,1].$ Then, she compares $d$ and $f_{k}(x),$ if
$d\leqslant f_{k}(x)$, she accepts $x$ as a non-Gaussian distribution data
after virtual subtraction of $k$ photons. Otherwise, she rejects it and
restarts the above process until all the raw data are completed.

After completing the non-Gaussian postselection process, they use the saved
data to perform postprocessing process through classical authentication
channel, including information reconciliation, parameter estimation, and
privacy amplification. The secret key rate against collective attacks for
reverse reconciliation of the $k$ photons subtraction is described by

$K_{PS}^{k}=P(k)[\beta I(A:B)-S(E:B)],$ (4)

where $P(k)$ is the success probability of overall virtual $k$ photons
subtraction (the success probability of the non-Gaussian postselection),
$\beta$ is reconciliation efficiency, $I(A:B)$ is classical mutual information
between Alice and Bob, $S(E:B)$ is Von Neumann entropy between Eve and Bob.

###  II.2 Non-Gaussian reconciliation in CV-QKD

Suppose that the variables of Alice and Bob are $X$ and $Y$ after non-Gaussian
postselection. $X$ follows non-Gaussian distribution and $Y$ still follows
Gaussian distribution. The quantum channel can be seen as an additive white
Gaussian noise (AWGN) channel. Then we have $Y=tX+Z$, where $t$ is related to
the channel transmittance and detection efficiency, $Z$ is channel noise which
follows Gaussian distribution. For information reconciliation, signal to noise
ratio (SNR) is the main parameter of concern. Thus for simplicity, we can fit
$t=1$. In addition, direct reconciliation is limited by 3dB loss. Therefore,
reverse reconciliation is used in our scheme, which can break this limit.
Alice performs the error correction to obtain the identical keys with Bob.

Alice and Bob first convert the AWGN channel to a virtual binary input AWGN
channel. Then they can use channel coding and decoding technology to correct
errors between them. Multidimensional reconciliation is used to finish the
first step. Bob normalizes the data $Y$ after postselection according to the
dimension of multidimensional reconciliation. Then he randomly chooses a
uniform distribution vector $U$ which is generated by a quantum random number
generator. Next, a mapping function $M(Y^{\prime},U)$ from normalized variable
$Y^{\prime}$ to $U$ is constructed by orthogonal transformation matrix.
$Y^{\prime}$ and $U$ satisfy the following relationship

$M(Y^{\prime},U)Y^{\prime}=U,$ (5)

He sends the mapping function to Alice through a classical authentication
channel which means that eavesdropper Eve can get all the information but she
can not change the information without the knowledge of both sides of the
legal communication. Alice receives the mapping function $M(Y^{\prime},U)$ and
normalizes her non-Gaussian distribution data $X$ after postselection. Then
she rotates her normalized data $X^{\prime}$ to $V$ through the mapping
function, which can be calculated by

$M(Y^{\prime},U)X^{\prime}=V,$ (6)

where $V$ is the noise form of $U$. They repeat the above process until all
the data are converted. Finally, they select the appropriate channel codes
according to the channel parameters to perform the error correction process.

MET-LDPC codes are the generalization form of LDPC codes, which is very
suitable for error correction under extremely low SNRs. They can achieve
performance close to the Shannon’s limit. Thus, we choose MET-LDPC code as the
channel coding technology to correct the errors between Alice and Bob.
Firstly, The code rate of MET-LDPC code is calculated according to the
estimated quantum channel characteristics. Secondly, the degree distribution
of the code rate is obtained by density evolutionary algorithm. Then select a
suitable construction method to generate the parity check matrix. Finally,
Alice and Bob use the matrix for encoding and decoding to correct the errors
between them to get completely consistent data.

The encoding process in CV-QKD system is very different from that in classical
communication. It does not need generation matrix. The parity check matrix of
error correction code is directly multiplied by the binary data after
reconciliation to obtain the syndromes. Then Bob sends the syndromes to Alice
through a classical authentication channel which will be errorless in the
transmission process but Eve can get all the information. This is different
from classical communication. After getting the syndromes, Alice initializes
the information according to the data after reconciliation and uses the same
parity check matrix to correct errors. In the message iteration process of
error correction, the syndromes sent by Bob needs to be used and compared with
Alice’s temporary syndromes to judge whether the decoding is successful. This
is another big difference from classical communication.

The error correction codes used in CV-QKD are MET-LDPC codes, which have many
decoding algorithms, such as probability domain belief propagation algorithm
(BP), log likelihood ratio belief propagation algorithm (LLR-BP) and so on, in
which BP algorithm is a commonly used decoding algorithm in CV-QKD system.
Compared with other decoding algorithms, it has higher accuracy and can ensure
the success rate of decoding in CV-QKD system. In the conventional BP decoding
algorithm, after information initialization, it is necessary to traverse the
check nodes and variable nodes to update the edge information in the parity
check matrix, then updating the information of each node accordingly, finally
comparing the syndromes. If the decision is successful, end the decoding,
otherwise continue the iteration until the maximum number of iterations is
reached. Although this algorithm has high accuracy, the updated nodes are in a
waiting state before the next update, the utilization rate is low. Thus, a
layered BP (LBP) decoding algorithm [40] is introduced to the postprocessing
of CV-QKD system, which can make faster use of the updated node information
and improve the decoding efficiency. In LBP decoding algorithm, each layer
will update the variable node, taking the updated variable node as the input
of the next layer to participate in the operation of the next layer in the
same iteration. In this way, the updated information will be immediately used
in this iteration, which can reduce the iteration number. Generally, it only
needs half of the iteration number to achieve the same effect as the BP
decoding algorithm. Thus, it can speed up the decoding process and it is
suitable for application in high-speed postprocessing for CV-QKD system.

##  III Performance of the protocol

In this section, we first present the performance of the non-Gaussian
postselection in terms of sampling efficiency from Gaussian distribution data
to Non-Gaussian distribution data. Then we present the performance of non-
Gaussian reconciliation for CV-QKD system in terms of reconciliation
efficiency, frame error rate of error correction, and average iteration
number.

###  III.1 Non-Gaussian postselection performance

As described in Sec. II, for the virtual $k$-photon subtraction, we cannot
directly sample non-Gaussian data from the raw data of Gaussian distribution
due to the probability of Gaussian distribution may be higher than that of
non-Gaussian distribution. We present the acceptance-rejection sampling method
to solve this problem, in which the sampling efficiency is a very important
parameter to evaluate sampling performance. In addition to Eq. 3, the sampling
efficiency can also be expressed by geometric interpretation. It refers to the
probability that the data conforming to $c_{k}g(x)$ falls under $f_{k}(x)$.

The selection of $c$ in Eq. 2 has an important impact on sampling efficiency.
First, it needs to satisfy that $f_{k}(x)$ is completely below $cg(x)$ in
order to sample correctly. Secondly, the sampling efficiency should be as high
as possible. That is, the value of $c_{k}$ should be as small as possible when
condition 1 is met. Therefore, its value can be calculated by

$c_{k}=\max_{x}[\dfrac{f_{k}(x)}{g(x)}],x\in R.$ (7)

As shown in Fig. 3, we give the Gaussian distribution probability density
function of actual data and the corresponding non-Gaussian probability density
function with virtual subtraction of one photon. We can obtain that the
optimal value of $c_{1}$ is about 1.32 calculated by Eq. 7 when $T$ is set to
0.8.

Figure 3: The Gaussian distribution probability density function of actual
data and non-Gaussian probability density function with virtual subtraction of
one photon. The thin black line represents the probability density of raw data
actually generated. The thin red line represents the probability density of
postselected data after virtual subtraction of one photon. $g(x)$ and
$f_{1}(x)$ are fitted probability density functions of raw data and
postselected data (virtual subtraction of one photon) separately. The pink dot
represents the point where $c$ gets the optimal value when virtual subtracting
one photon, recording as $c_{1}$. The purple line represents the probability
density function after $g(x)$ is enlarged by $c_{1}$ times. The modulation
variance is set to 20.

The corresponding sampling efficiency is about 75.4%, which is much higher
than the previous results of sampling with uniformly distributed data.
Similarly, we can get the results of virtual $k$-photon subtraction.

###  III.2 Non-Gaussian reconciliation performance

Information reconciliation has an important impact on the performance of CV-
QKD system. Reconciliation efficiency not only affects that whether secret
keys can be extracted, but also affects the transmission distance of CV-QKD
system. On the other hand, the success rate of reconciliation and processing
speed also have an important impact on the secret key rate of the system.
Thus, we test the error correction performance of both Gaussian and non-
Gaussian data, including reconciliation efficiency, FER and average iteration
number (AIN).

Figure 4: Performance comparison of error correction between Gaussian and non-
Gaussian data. The error correction code used for Gaussian and non-Gaussian
case is MET-LDPC code, whose rate is 0.1. (a) FER of Gaussian and non-Gaussian
data after error correction under different SNR. (b) Average iteration number
(AIN) of decoding corresponding to Gaussian and non-Gaussian data in (a). The
blue line represents the FER/AIN of Gaussian data after error correction. The
rest lines represent the FER/AIN of non-Gaussian data, where red line/yellow
line/purple line respectively represents virtual 1-photon/2-photon/3-photon
subtraction (VPS-1/VPS21/VPS-3). The dots represent the error correction
performance of actual data, reconciliation efficiency from right to left is
93%, 93.5%, 94%, 94.5%, 95%, 95.5%, 96% respectively. The maximum iteration
number is set to 150. Figure 5: Performance comparison of error correction
between Gaussian and non-Gaussian data under the code rate of 0.05. (a) FER of
Gaussian and non-Gaussian data after error correction under different SNR. (b)
Average iteration number (AIN) of decoding corresponding to Gaussian and non-
Gaussian data in (a). The dots represent the error correction performance of
actual data, reconciliation efficiency from right to left is 93.5%, 94%,
94.5%, 95%, 95.5%, 96%, 96.5% respectively. The maximum iteration number is
set to 150.

The error correction performance of four types data of distribution is tested
under two MET-LDPC codes. The data of these four types of distribution
contains Gaussian distribution data and three non-Gaussian distribution data
which includes the data of virtual 1-photon subtraction, virtual 2-photon
subtraction, virtual 3-photon subtraction. The two MET-LDPC codes are rate of
0.1 and 0.05. For each rate of MET-LDPC code and each type of data, seven sets
of data with different SNR are tested. The test results are shown in Fig. 4
and Fig. 5.

We have tested more than 100 data block for each case. The size of each data
block is $10^{6}$. As can be seen from the results of Fig. 4 and Fig. 5, the
error correction performance of non-Gaussian case is higher than that of
Gaussian case both in FER and AIN. LBP decoding algorithm is used to the error
correction process, which has little effect on FER of decoding. However it can
reduce about half of the AIN, the decoding speed can be greatly increased to
improve the secret key rate of CV-QKD system. FER is related to the set
maximum number of iterations, the FER can be reduced by increasing the maximum
number of iterations to a certain extent. But it will also increase the
decoding delay simultaneously, thus there is a trade-off between FER and AIN.
In order to compare the error correction performance under different
conditions, we set the maximum number of iterations to 150 for all the tests.
In actual applications, the maximum number of iterations can be reasonably set
according to the average number of iterations for different cases.

Fig. 4 and Fig. 5 show the test results for two MET-LPDC codes, the code rate
of 0.1 and 0.05 separately. Although the error correction performance of non-
Gaussian data is better than that of Gaussian data at both codes, it also have
some different characteristics. For the code rate of 0.1, the error correction
performance of non-Gaussian data is obviously better than that of Gaussian
data both in FER and AIN, the error correction performance is better with the
increase of the number of virtual photon subtraction, especially at a
relatively low SNR. This is mainly due to the fact that non-Gaussian
postselection diffuses the data originally concentrated around 0 to middle
values, the larger the values diffusion with the increase of the number of
virtual photon subtraction. Simultaneously, the large value will also
concentrate to the middle value. Therefore, the error correction performance
will not continue to increase. At a relatively high SNR, error correction is
relatively easy at this time, so FER of error correction is very low (close to
0) for both Gaussian data and non-Gaussian data, continuing to increase the
SNR cannot reflect the advantages of non-Gaussian data. However, non-Gaussian
data error correction can converge faster, so the AIV is less than that of
Gaussian data which will increase the error correction speed and the secret
key rate of CV-QKD system.

For the code rate of 0.05, the error correction performance of non-Gaussian
data is also higher than that of Gaussian data, but its advantage is lower
than that of 0.1 code rate matrix. This is because the SNR of error correction
data corresponding to 0.05 code rate is low, the power of signal is much
smaller than that of the noise. In this case, the signal is completely
submerged in the noise, so it is difficult to correct the errors. Therefore,
the advantages of non-Gaussian postselection are limited, with the increase of
the number of photon subtraction, its advantage is not as obvious as that in
0.1 code rate. Although the error correction gain caused by different photon
subtraction is not very obvious, the non-Gaussian postselection data still
makes the decoding converge faster than that of Gaussian data due to the
reduction of the values near 0. Therefore, the FER and AIN performance of
decoding processing is still improved.

We can use rate-adaptive method [17] or non-fixed rate error correction codes
[28] to expand the applicable range of SNR. We have studied these two methods
in the preamble work, combining these two methods, the non-Gaussian error
correction method proposed in this paper can play an advantage in a large
range of SNR.

Figure 6: The secret key rate and transmission distance comparison of the
original protocols (Gaussian case) and 1-photon subtraction (non-Gaussian
case). The modulation variance is 20. Reconciliation efficiency is 95%.

Fig. 6 shows the secret key rate and transmission distance comparison of the
original protocol (Gaussian case) and 1-photon subtraction protocol (non-
Gaussian case). Fig. 6 shows that non-Gaussian protocols can still achieve
high secret key rate over long distance range, which greatly expanding the
maximal transmission distance. However, for short distance range, the secret
key rate is worth than the original protocols. The main reason is that the
probability of photon subtraction success is low. In other words, for non-
Gaussian postselection, after selecting the original Gaussian data, since the
amount of selected data is reduced, the average secret key rate is reduced.

In practical application, we are more concerned about the amount of secret
keys obtained per unit time when the secret key rate is greater than 0. For
non-Gaussian postselection, the reduction of data will lead to the reduction
of the secret key rate of single pulse, but it will greatly improve the data
processing speed. The reconciliation efficiency and decoding success rate of
non-Gaussian data are higher than that of Gaussian data, thus the secret key
rate per unit time is not necessarily lower than that in Gaussian case even at
short distance range. It can be further studied in the subsequent high-speed
implementation.

##  IV Conclusion

In this paper, we proposed a non-Gaussian reconciliation method for CV-QKD
protocols by non-Gaussian postselection at Alice’s side, which can reduce the
FER and AIN of decoding. We propose an effective postselection method to
obtain non-Gaussian data follows specific distribution from Gaussian data,
which greatly improves the success rate of virtual photon subtraction of CV-
QKD system. Multidimensional reconciliation and MET-LDPC codes are used to
perform the information reconciliation in postprocessing of CV-QKD system. The
layered belief propagation decoding algorithm is introduced to the error
correction, which can greatly reduce the decoding complexity and improve the
decoding speed. We test the error correction performance of Gaussian data and
non-Gaussian data after our proposed postselection under two representative
codes with rate of 0.1 and 0.05. We test 7 sets of data for each case
respectively. The corresponding reconciliation efficiency ranges from 93% to
96.5%. The results show that the FER and AIN of decoding performance of non-
Gaussian data is significantly better than that of Gaussian data, which
greatly improves the secret key rate of CV-QKD system.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China under
Grant No. 62001041, No.62201012, the Fundamental Research Funds of BUPT under
Grant No. 2022RC08, and the Fund of State Key Laboratory of Information
Photonics and Optical Communications under Grant No. IPOC2022ZT09.

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