High-rate discretely-modulated continuous-variable quantum key distribution
using quantum machine learning

In general, the whole process of one-way CVQKD protocol includes five steps, i.e., state preparation, measurement, parameter estimation, data reconciliation and privacy amplification. Figure 1 shows the process of conventional CVQKD, and each step is explained as follows.

State Preparation. Alice prepares a train of coherent states whose quadrature values $x$ and $p$ obey a bivariate Gaussian distribution. Then the modulated coherent states are sent to Bob through an untrusted quantum channel.

Measurement. The pulses sent by Alice are received by Bob who measures these incoming signal states with coherent detector. After enough rounds, a string of raw key can be shared between Alice and Bob.

Parameter Estimation. Alice and Bob disclose part of the raw key to calculate the transmittance $T$ and excess noise $\varepsilon$ of the quantum channel. With these two parameters, the secret key rate can be estimated.

Data Reconciliation. If the estimated secret key rate is positive, the low density parity check (LDPC) code is applied to another part of the raw key, aiming to error correction.

Privacy Amplification. Finally, a privacy amplification algorithm is performed based on the hash function, so as to extract the final secret key that is entirely unknown to the eavesdropper Eve.

$k$NN is a traditional lazy learning algorithm that uses a majority vote to classify (predict) the grouping of unlabeled data points. For an unlabeled data point $v_{0}$ and a training set containing $v_{j}\;(j=1,2,...,M)$ labeled data points, $k$NN first finds a set of labeled data points whose similarities with $v_{0}$ are the top $k$ highest among all data pints in training set, and then counts the number of each label, the label with biggest number will be assigned as the label of data point $v_{0}$. Figure 2 depicts an example of $k$NN algorithm in 2-dimensional feature space, in which a gray dot denotes an unlabeled data point and it is going to be labeled as green or blue. The gray dot will be assigned to the blue class when $k=3$ due to there are two of the three-nearest labeled data points belong to the blue class while only one point belongs to the green class. Similarly, it will be labeled as green class for $k=7$ as there are four of the seven-nearest labeled data points belong to the green class while only three point belongs to the blue class.

The specific steps of $k$NN can be seen in Table 1. It is worth noting that the similarity is one of the crucial parameters of $k$NN algorithm. Assuming ${\bf v_{0}}=(v_{01},v_{02},...,v_{0U})$ and ${\bf v_{j}}=(v_{j1},v_{j2},...,v_{jU})$ are the respective feature vectors of unlabeled data point $v_{0}$ and labeled data point $v_{j}$, similarity can be measured by following ways.

Euclidean distance

Cosine similarity

In the above two similarities, Euclidean distance counts the absolute distance of each data point, revealing the difference of each data point’s location in the coordinate system Van Der Heijden et al. (2005). Cosine similarity counts the cosine of an angle between two vectors, revealing the directional difference of each vector Tan et al. (2016). In addition, Manhattan distance Stigler (1986), Hamming distance Waggener et al. (1995) and inner product also can be used as similarity in $k$NN, one should select a proper way according to different applications.

As we mentioned above, Bob needs to prepare quantum states ($|\tau\rangle$ or $|\rho\rangle$) in both initialization part and quantum computing processing part. The difference is that quantum state $|\tau\rangle$ is prepared in initialization part to carry the information of all feature vectors that extracted from labeled DMCSs, while quantum states $|\rho\rangle$ is prepared in prediction part to carry the information of feature vector that extracted from an unlabeled DMCS. Assuming the normalized feature vectors that extracted from labeled DMCSs are $V=\{{\bf v_{1}},{\bf v_{2}},...,{\bf v_{M}}\}$ and the normalized feature vector that extracted from an unlabeled DMCS is ${\bf v_{0}}$, the quantum states $|\tau\rangle$ and $|\rho\rangle$ can be respectively expressed by Wiebe et al. (2014)

and

where $v_{ji}$ $(v_{0i})$ is the $i$-th feature value of feature vector ${\bf v_{j}}$ $({\bf v_{0}})$, $U$ is the dimension of the extracted features ($U=8$ in our case) and $M$ is the number of labeled DMCSs. Obviously, $|\tau\rangle$ is the superposition state that carries the information of all feature vectors for known (labeled) DMCSs, while $|\rho\rangle$ is the quantum state that carries the information of feature vector for an unknown (unlabeled) DMCS.

In what follows, we show how to obtain these two quantum states. To prepare quantum state $|\tau\rangle$, we first need to prepare an initial state $|\tau_{init}\rangle$ which can be expressed as

where state $\frac{1}{\sqrt{M}}\sum_{j=1}^{M}|j\rangle$ (or state $\frac{1}{\sqrt{U}}\sum_{i=1}^{U}|i\rangle$) can be obtained by a quantum circuit shown in Fig. 6. See Appendix A for the detailed derivation of initial state $|\tau_{init}\rangle$. Subsequently, an Oracle ${\mathcal{O}}|j\rangle|i\rangle|0\rangle=|j\rangle|i\rangle|v_{ji}\rangle$ is applied to this initial state so that the resultant state $|\tau_{o}\rangle$ can be expressed by

After that, an unitary operation

is applied to the last quantum bit (qubit) of $|\tau_{o}\rangle$, so that the rotated state

can be obtained. Finally, the quantum state $|\tau\rangle$ can be obtained by removing the auxiliary qubit $|v_{ji}\rangle$ of $|\tau_{r}\rangle$ using Oracle ${\mathcal{O}}^{\dagger}$. Similarly, the quantum state $|\rho\rangle$ can be prepared in the same way with initial state $|\rho_{init}\rangle=\frac{1}{\sqrt{U}}\sum_{i=1}^{U}|i\rangle|0\rangle|0\rangle$. As a result, the information of feature vectors can be stored in the amplitude of quantum states $|\tau\rangle$ and $|\rho\rangle$.

For now, Bob possesses both the quantum state $|\tau\rangle$ that carries the information of known DMCSs and the quantum state $|\rho\rangle$ that carries the information of unknown DMCS. He first calculates the fidelity, which can be used as a measure of the similarity, between these quantum states and store it to the amplitude of qubit by performing a controlled-SWAP (c-SWAP) test shown in Fig. 7. The quantum circuit of c-SWAP test is composed of two Hadamard gate and a SWAP operation which obeys ${\rm SWAP}|\rho\rangle|\tau_{j}\rangle=|\tau_{j}\rangle|\rho\rangle$, where

Therefore, the function of c-SWAP test is that the last two states will be swapped if the control qubit is $|1\rangle$ or they will not be swapped if the control qubit is $|0\rangle$.

After passing the c-SWAP test, the input state can be transformed to

where

See Appendix B for the detailed derivation of $P_{j}(0)$. It is easy to find that $P_{j}(0)$ is directly proportional to the fidelity $|\langle\rho|\tau_{j}\rangle|^{2}$, so that $P_{j}(0)$ can be directly used for measuring the similarity between $|\rho\rangle$ and $|\tau_{j}\rangle$. After $M$ times c-SWAP test, a superposition state whose amplitudes contain all similarities can be finally obtained as

In order to facilitate the processing of follow-up quantum search algorithm, amplitude estimation Brassard et al. (2002) is further applied to this superposition state so that the amplitudes of $|\gamma\rangle$ can be stored as a qubit string. Figure 8 shows the quantum circuit for implementing amplitude estimation, which includes two steps i.e., amplitude amplification and phase estimation. Specifically, amplitude amplification is implemented by an unitary operator $Q=-{\mathcal{A}}S_{0}{\mathcal{A}}^{\dagger}S_{\chi}$ where $\mathcal{A}$ performs $|0^{\otimes m}\rangle\rightarrow|\gamma\rangle$, $S_{0}$ obeys

and $S_{\chi}$ obeys

while phase estimation is implemented by an inverse quantum Fourier transform (IQFT) ${\mathcal{F}}^{-1}$ which is defined as

where $0\leq x<M$, $\iota=\sqrt{-1}$. After amplitude estimation, a quantum state

can be prepared, where ${\rm Sim}_{j}=\frac{M}{\pi}{\rm arcsin}(\sqrt{\widetilde{P_{j}}(0)})$ and $\widetilde{P_{j}}(0)$ is the estimate of $P_{j}(0)$. See Appendix C for the detailed derivation of quantum state $|\sigma\rangle$. For now, the similarities can be deemed to have been stored as a qubit string due to ${\rm Sim}_{j}$ is also directly proportional to the fidelity $|\langle\rho|\tau_{j}\rangle|^{2}$.

Bob now holds the quantum state $|\sigma\rangle$ whose qubit string contains the information of similarities between all known DMCSs and the unknown DMCS. He then finds $k$ nearest neighbors of the unknown DMCS by applying quantum $k$-maximal finding algorithm, which is detailed as follows.

As shown in Fig. 9, Bob first randomly selects $k$ known DMCSs and records their corresponding indexes $j$ into an initial set $K$, the rest indexes $j^{\prime}$ are recorded into a set $K^{\prime}$. Then, the Grover’s algorithm Grover (1996) is applied to $|\sigma\rangle$ to find an index $j^{\prime}$ whose corresponding $|{\rm Sim}_{j^{\prime}}^{K^{\prime}}\rangle$ satisfying ${\rm Sim}_{j^{\prime}}^{K^{\prime}}>{\rm Sim}_{j^{*}}^{K}$, where $j^{*}\in K$. In our case, the quantum circuit for implementing Grover’s algorithm is shown in Fig. 10, which includes a number of Grover iterations and a final measurement. Each iteration is composed of a unitary operator $S_{F}$ and a Grover diffusion operator ${\mathcal{D}}$. Specifically, $S_{F}$ denotes conditional phase shift transformation which obeys

where the function $F$ is defined as

The $S_{F}$ operator can be implemented by a CMP operator and a CNOT gate. After passing the CMP operation, the fourth input qubit $|0\rangle$ is transformed to

Then, the CNOT gate is controlled by the above state so that the last input qubit $|0\rangle$ is transformed to

Therefore, the resultant state

can be obtained after applying $S_{F}$ operator. The Grover diffusion operator ${\mathcal{D}}=W^{\prime}S_{0}W^{\prime\dagger}$ is subsequently applied to the first qubit of the resultant state, so that its amplitude will be inversed about average Grover (1997). After the first Grover iteration, the state

where $t$ is the number of solutions of Grover’s algorithm, can be obtained. From Eq. (21) and Eq. (22), we can easily find that the amplitude of $\sum_{F(j)=1}|j\rangle$ is amplified. After enough rounds of Grover iteration, the index $j^{\prime}$ can be obtained by the final measurement with a probability approaching 1 (Details about this iteration is presented in Appendix D). Bob then replaces $j^{*}$ with $j^{\prime}$, and starts a new round of Grover’s algorithm. Finally, the indexes of $k$ nearest neighbors can be obtained from the final set $K$.

As the proposed Q$k$NN can be deemed a quantum version of $k$NN algorithm, classical machine learning metrics can be used for evaluating its performance. In machine learning area, the confusion matrix is one of the most widely used concepts to analyze the performance of a certain classification algorithm. Table 2 shows the confusion matrix, in which the true positive ($\mathcal{N}_{TP}$) indicates the counts that actual positive data are predicted as positive, false positive ($\mathcal{N}_{FP}$) indicates the counts that actual negative data are predicted as positive, false negative ($\mathcal{N}_{FN}$) indicates the counts that actual positive data are predicted as negative, and true negative ($\mathcal{N}_{TN}$) indicates the counts that actual negative data are predicted as negative.

With confusion matrix, a machine learning metric called Precision ($\mathcal{P}_{\rm Prec}$) can be defined as

which indicates the proportion of actual positive data in all predicted positive data. This metric is important to our case as raw key is generated from the correct labels in all the predicted labels. Since there are eight labels existed in phase space, we further calculate the precision of each label $\mathcal{P}_{{\rm Prec}_{i}}$ and average them to obtain a metric Average Precision $\bar{\mathcal{P}}_{\rm Prec}$, i.e.

Obviously, the higher the value of $\bar{\mathcal{P}}_{\rm Prec}$, the more correlated the raw key. Figure 11 shows the average precision $\bar{\mathcal{P}}_{\rm Prec}$ of the proposed Q$k$NN algorithm with different hyper-parameter $k$. By and large, it can be found that the value of $\bar{\mathcal{P}}_{\rm Prec}$ decreases with transmission distance increases, which indicates that the transmission distance is a crucial factor that impacts the performance of Q$k$NN algorithm. This is in line with our expectation, as the increased transmission distance will lead to more channel losses, resulting in extremely lower SNR of the received DMCSs. In addition, we find that $\bar{\mathcal{P}}_{\rm Prec}$ fluctuates as the hyper-parameter $k$ varies, and this situation exists in all distances. To select a proper $k$, we mark the peak value of each line out with a circle. It is observed that these peak values occur when $k$ is set from 11 to 17, e.g. $\bar{\mathcal{P}}_{\rm Prec}=0.9541$ at transmission distance of $5$ km when $k=15$, which suggests that the value of $k$ should not be set too small or too large. This is because a small value of $k$ may lead to overfitting problem and a large value of $k$ may result in underfitting problem, while $\bar{\mathcal{P}}_{\rm Prec}$ will be decreased by either of the two problems Hastie et al. (2001).

For now, we have investigated the performance of Q$k$NN in terms of precision. To comprehensively evaluate a classifier, however, a sole metric is inadequate. Therefore, an overall metric called macro-average Receiver Operating Characteristic (ROC) curve Fawcett (2006), which describes the average True Positive Rate ($\mathcal{P}_{\rm TPR}$) of a certain multi-class classifier as a function of its average False Positive Rate ($\mathcal{P}_{\rm FPR}$), is introduced. For each class, $\mathcal{P}_{\rm TPR}$ indicates the proportion of the data that is correctly predicted as positive in all actual positive data, and $\mathcal{P}_{\rm FPR}$ indicates the proportion of the data that is incorrectly predicted as positive in all actual negative data. Their formulas are given by

Figure 12 shows the macro-average ROC curves of the proposed Q$k$NN with hyper-parameter $k=15$. The dashed line is the result of random guess, which illustrates that there is no performance improvement without using any classification algorithm. With macro-average ROC curve, one can explicitly tell the quality of the multi-class classifier: the curve more close to point $(0,1)$, the performance better. Obviously, the proposed Q$k$NN can dramatically improve the performance of predicting DMCSs. Moreover, it can be observed that these curves are away from the best point $(0,1)$ with the transmission distance increases, this trend is identical with our previous analysis, which further illustrates that channel loss is important for the performance of Q$k$NN algorithm. Besides, we further calculate the area under curve (AUC) Bradley (1997) for each distance and thus obtain AUC value $\Lambda_{\rm Q}$, which is a probability value range from 0 to 1. As a numerical value, $\Lambda_{\rm Q}$ can be directly used for quantitatively evaluating classifier’s quality. It can be easily found that our proposed Q$k$NN algorithm achieves extremely high overall performance ($\Lambda_{\rm Q}=0.9946$) with transmission distance is $5$ km. It is worth noting that the AUC value of random guess is 0.5, while it gets 0.8016 with the proposed Q$k$NN even the transmission distance is increased to $50$ km. This result demonstrates that the effectiveness of Q$k$NN in CVQKD system, therefore, the AUC value $\Lambda_{\rm Q}$ can be used to describe the efficiency of quantum classifier in the following security analysis of the proposed Q$k$NN-based CVQKD system.

Before presenting the complexity analysis of Q$k$NN, let us briefly retrospect the complexity of classical $k$NN first. Assuming there are $M$ training data points $V_{j}=\{{\bf v_{1}},{\bf v_{2}},...,{\bf v_{M}}\}$ and each training data point is represented as an $U$ dimensional feature vector ${\bf v_{j}}=(v_{j1},v_{j2},...,v_{jU})$, the classical $k$NN algorithm first needs to calculate all similarities between the unlabeled data point ${\bf v_{0}}$ and each labeled data point ${\bf v_{j}}$, so that the complexity of this step is $O(UM)$. Then, the $M$ similarities need to be sorted with a certain sorting algorithm such as quick sort, merge sort, heap sort, etc. In general, the complexity of above sorting algorithms is no less than $O(M{\rm log}_{2}M)$ Cormen et al. (2022). Finally, the majority voting requires $O(k)$ times to count the number of labels and assigns the label to ${\bf v_{0}}$. As a result, the total complexity of classical $k$NN can be expressed by

In what follows, let us discuss the complexity of the proposed Q$k$NN algorithm in detail. As shown in Fig. 5, the whole Q$k$NN is composed of four parts, i.e., preparation of quantum states, similarity calculating, quantum $k$-maximal finding and majority voting. For preparation of quantum states, as we detailed in Sec. III.1, both quantum states $|\tau\rangle$ and $|\rho\rangle$ are prepared by 3 Oracles (i.e. $\mathcal{O}$, $R_{y}$ and $\mathcal{O}^{\dagger}$ ), so that the complexity of this part is $O(6)$. For similarity calculating, the two quantum states $|\tau\rangle$ and $|\rho\rangle$ are first passed through the c-SWAP test quantum circuit to obtain the quantum state $|\gamma_{j}\rangle$ that contains the similarity $P_{j}(0)$. Note that the complexity of calculating a $P_{j}(0)$ is $O({\rm log}^{2}_{2}U)$ Hai et al. (2022). To obtain the superposition state $|\gamma\rangle$ whose amplitudes contain all similarities, the c-SWAP test needs to be performed $M$ times, the complexity is thereby increased to $O(M{\rm log}^{2}_{2}U)$. Subsequently, the similarities need to be stored as a qubit string by amplitude estimation. Specifically, the $Q$ operator is repeatedly performed to estimate the amplitude $a$ (see details in Appendix C), and the error probability for estimating $a$ satisfies Brassard et al. (2002)

where $\widetilde{a}$ is the estimation of $a$, and $R$ is the iteration times of operator $Q$. Obviously, $R$ needs to satisfy the following inequation, i.e.

if the error probability $|a-\widetilde{a}|\leqslant\delta$. That is to say, the $Q$ operator needs to be performed at least $R$ times ($O(R)$) to ensure that the error probability is less than or equal to $\delta$. Therefore, the total complexity of similarity calculating is $O(M{\rm log}^{2}_{2}U+R)$. For quantum $k$-maximal finding, let ${\mathcal{T}}$ be a set whose elements do not belong to set $K$ but are more similar to ${\bf v_{0}}$ than some points in set $K$, and the number of elements of set ${\mathcal{T}}$ is $t$. Obviously, to find out $k$ maximal values, the Grover’s algorithm and replacement need to be repeatedly performed until set ${\mathcal{T}}$ is empty, i.e., $t=0$. Reference Durr et al. (2006) shows that $t$ can be reduced to $\frac{t}{2}$ by performing $O(k)$ iterations of Grover’s algorithm and replacement when $t>2k$, i.e., the Oracle needs to preform $O(k\sqrt{\frac{M}{t}})$ times. Once $t$ is reduced to $t\leq 2k$, $O(\sqrt{\frac{M}{t}})$ times Oracles in each round of Grover’s algorithm are required to ensure $t=0$. Therefore,

times Oracles are required for reducing $t$ to 0. From Eq. (30), we can easily find that the complexity of quantum $k$-maximal finding is $O(\sqrt{kM})$. Till now, we have presented the complexity analysis of the former three parts, note that the last part of Q$k$NN, namely the majority voting, is similar to that of classical $k$NN, so that its complexity remains $O(k)$. Therefore, the total complexity of our proposed Q$k$NN is

Figure 13 shows complexity comparisons between the proposed Q$k$NN and classical $k$NN. As can be seen in Fig. 13(a), the total complexity of the proposed Q$k$NN is much less than that of classical $k$NN, it illustrates that the proposed Q$k$NN algorithm could offer a significant speedup over classical $k$NN algorithm. To figure out how the acceleration happens, we further mark each part with different colors and the result shows that the complexities of similarity calculating and $k$-maximal finding (corresponding to sorting in classical $k$NN algorithm) are dramatically decreased by the proposed Q$k$NN. It illustrates that the quantum parts of Q$k$NN are crucial for speedup. In addition, we find that the complexities for both Q$k$NN and classical $k$NN algorithms are affected by several parameters, i.e., the dimension $U$, the number of training data $M$ and the number of nearest neighbors $k$. We therefore plot Fig. 13(b), Fig. 13(c) and Fig. 13(d) to investigate the respective influence of each parameter on complexity. As shown in Fig. 13(b) and Fig. 13(c), the complexity of classical $k$NN rises quite sharply with the increase of $U$ or $M$, while the complexity of Q$k$NN rises very slowly. It illustrates that the proposed Q$k$NN algorithm is of clear superiority in addressing high dimensional or large-size classification issues. Figure 13(d) shows that although the complexity of classical $k$NN is apparently larger than that of Q$k$NN, the complexities of both algorithms are not sensitive to the hyper-parameter $k$, which illustrates that $k$ is not a crucial parameter that heavily affects the complexities of both classical $k$NN and Q$k$NN algorithms. It is worthy noting that, to explicitly show the respective trends, the scale of labeled data point $M$ is set to 128 in Fig. 13(a), Fig. 13(b) and Fig. 13(d). Actually, $M$ is usually far more than $10^{5}$ for a realistic CVQKD system Leverrier (2015). In such a practical scenario, the complexity gap between Q$k$NN and classical $k$NN will be extremely large.

Till now, we have demonstrated the performance of Q$k$NN-based CVQKD in terms of machine learning metrics and have analyzed the complexity of Q$k$NN algorithm, both results have shown the advantages of our scheme. In what follows, we present the theoretical security proof for Q$k$NN-based CVQKD with semi-definite program (SDP) method Denys et al. (2021), detailed calculations can be found in Appendix E. As known, the asymptotic secret key rate of the conventional DM CVQKD with reverse reconciliation is given by Liao et al. (2020b)

where $\beta$ is the reconciliation efficiency, $I_{\rm AB}$ is the Shannon mutual information between Alice and Bob, and $\chi_{\rm BE}$ is the Holevo bound of the mutual information between Eve and Bob. However, Eq. (32) does not consider the influence of the introduction of Q$k$NN classifier for both legitimate users (Alice and Bob) and the eavesdropper (Eve), it, therefore, has to be amended to suitable for evaluating quantum machine learning-based CVQKD. Due to the data processing is quite different, Eq. (32) can be rewritten as the following form

The difference between Eq. (32) and Eq. (33) lies in two parts. First, the AUC value $\Lambda_{\rm Q}$ has to be considered as it describes the efficiency of quantum classifier. Higher AUC value implies higher correlation of raw key between Alice and Bob. Second, term $\chi_{\rm BE}$, which represents the Holevo quantity for Eve’s maximum accessible information, is reduced to $p(y_{i})\chi_{\rm BE}$, where $p(y_{i})=1/N$ is the probability of discrete uniform distribution when variable $Y=y_{i}(i=1,2,...,N)$. This is because Eve is no longer able to acquire as much information as before, due to some relationships are no longer fixed and public. To be specific, in conventional DM CVQKD, the relationship between each DMCS and its binary presentation is fixed and public. For instance, the key bits of state $|\alpha_{0}\rangle$ in QPSK CVQKD is always (0, 0) (see Fig. 1 in Ref. Ghorai et al. (2019)), so that Eve can precisely recover the correct key bits (0, 0) when she successfully intercepts the coherent state $|\alpha_{0}\rangle$. Similarly for 8PSK CVQKD, Eve can precisely recover the correct key bits (0, 0, 0) when she successfully intercepts the coherent state $|\alpha_{0}\rangle$. However, due to the special-designed process of Q$k$NN-based CVQKD, the above relationship is no longer fixed and public. Although each DMCS is assigned to a fixed label, such as the label of $|\alpha_{0}\rangle$ is $L_{1}$ and the labels of $|\alpha_{7}\rangle$ is $L_{8}$ shown in Fig. 3, the binary presentation for each label can be randomly assigned by Alice. Since the initialization part is secure, Bob will learn the assignment by the transmitted DMCSs at the end of initialization. While Eve who does not participate the initialization part is completely unaware of the assignment shared by Alice and Bob. She thus can only guess the correct label with the success probability for $N$-PSK is $1/N$. As can be seen in Fig. 14, in our proposed scheme, Eve’s maximum accessible information is apparently reduced when compared with conventional DM CVQKD, and it can be further decreased by using higher dimensional PSK modulation. This is opposite to the trend in conventional DM CVQKD in which Eve’s maximum accessible information will increase with higher dimensional PSK modulation. It illustrates that the proposed scheme can efficiently prevent eavesdropper from obtaining more useful information, thereby contributing to the increase of final secret key.

Figure 15 shows the performance comparison between Q$k$NN-based CVQKD and two conventional DM CVQKD protocols in asymptotic limit. The results demonstrate that the secret key rate of Q$k$NN-based CVQKD outperforms other DM CVQKD protocols at all channel losses. In addition, we find that the secret key rate of the proposed scheme can be further increased with the risen modulation variance $V_{m}$. This is inconsistent with conventional DM CVQKD whose security has to be guaranteed by small modulation variance Ghorai et al. (2019); Liao et al. (2021). To investigate what caused this inconsistency, we plot Fig. 16, which shows the asymptotic secret key rates of above-mentioned schemes as a function of modulation variance. It can be easily found that the curves of both QPSK CVQKD and 8PSK CVQKD are arched and are located in certain ranges of small modulation variance, and the ranges are getting narrower with channel loss increases. Meanwhile, curves of Q$k$NN-based CVQKD are keep rising with the increase of modulation variance, it illustrates that small modulation variance is no longer needed for guaranteeing the security, so that the secret key rate of our proposed scheme can be further increased by setting proper larger modulation variance.