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A Survey of Machine Learning Assisted Continuous-Variable Quantum Key Distribution

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18 August 2023

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21 August 2023

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Abstract
Continuous-variable quantum key distribution (CV-QKD) shows potential for the rapid development of an information-theoretic secure global communications network; however, the complexities of CV-QKD implementation remain a restrictive factor. Machine learning (ML) has recently shown promise in alleviating these complexities. ML has been applied to almost every stage of CV-QKD protocols, including ML-assisted phase error estimation, excess noise estimation, state discrimination, parameter estimation and optimization, key sifting, information reconciliation, and key rate estimation. This survey provides a comprehensive analysis of the current literature on ML-assisted CV-QKD. In addition, the survey compares the ML algorithms assisting CV-QKD with the traditional algorithms they aim to augment, as well as providing recommendations for future directions for ML-assisted CV-QKD research.
Keywords: 
Subject: Engineering  -   Electrical and Electronic Engineering

1. Introduction

Quantum key distribution (QKD) utilises the principles of quantum mechanics to share a pair of secret keys between two parties to encrypt sensitive data with information-theoretic unconditional security (when using a one-time pad). In continuous-variable QKD (CV-QKD), information is normally encoded on the quadratures of an electric-field, which can be achieved using classical optical hardware, making its implementation simpler than the alternative discrete-variable QKD. For example, coherent measurement of CV quantum signals is possible using homodyne or heterodyne detectors [1].
Practical implementation of real-time CV-QKD is highly complex, with three major factors limiting its widespread implementation [2,3]. Firstly, signal distortion due to excess noise introduced by the channel and detector leads to loss of information, threatening to reduce secure key rates below the null key threshold. Secondly, in fluctuating channels, effective parameter estimation and optimization needs to be implemented if positive key rates are to be achieved. Thirdly, the computational time and power required to measure signals, filter out excess noise, and perform information reconciliation, reduce the real-time capability of CV-QKD.
While many traditional approaches have been suggested to overcome the limitations on CV-QKD, machine learning (ML) has recently been shown to have advantages in terms of phase error estimation and excess noise filtering [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19], state discrimination [20,21,22], parameter estimation and optimization [23,24,25], key sifting [26], reconciliation [27], and key rate estimation [28,29]. ML-based phase error estimation and noise filtering algorithms offer the potential of improved filtering capabilities due to their ability to map complex relationship between inputs and outputs based on the data alone, without being based on idealistic models which may not represent reality. Likewise, the ability of ML models to adapt to different system conditions, without reliance on assumptions-based traditional models, offer improved system parameter estimation and optimization capabilities. Further, traditional algorithms often rely on numerical searches to parameterize models, which can be more computationally complex than their ML-based counterparts.
This survey aims to provide a comprehensive overview of the current literature on the use of ML algorithms to assist in achieving practical CV-QKD. An emphasis is placed on comparing the performance of the ML algorithms when compared to their traditional counterparts when possible, such that the advantage of ML in assisting CV-QKD is highlighted.
In the following, ML algorithms adopted for different aspects of CV-QKD are contrasted with one another, and the difference in ML approaches to Gaussian modulated coherent state (GMCS) CV-QKD are compared with those of discrete-modulated (DM) CV-QKD. Beyond this, an analysis of the ML algorithms used for the two main channel types, free-space optical (FSO) and optical fiber, is undertaken, and how various common assumptions could affect the ML-based approaches is outlined.
Not included in this survey, is research on ML-assisted quantum attacks [30,31], attack detection and prevention [25,32,33,34,35,36,37,38], and methods for hacking ML-based attack prevention strategies [39,40]. Huang, Liu, and Zhang [41] have previously reviewed the literature in this area. We note that [31,35,36,37,38,39,40] were published after [41], so were not included in their review. It should also be made clear that none of the ML-assisted CV-QKD research described here involves quantum ML - relating to theoretical ML algorithms designed to run on quantum computers [42].
The rest of this paper is organized as follows: A brief overview of CV-QKD is provided in Section 2, followed by an introduction to ML techniques in Section 3. The application of ML to GMCS CV-QKD is outlined in Section 4, DM CV-QKD in Section 5, parameter estimation and optimization in Section 6, and key sifting, reconciliation, and key rate estimation in Section 7. A general discussion is given in Section 8, and potential future research directions are highlighted in Section 9. Concluding remarks are provided in Section 10.

2. CV-QKD Overview

CV-QKD involves two legitimate parties, Alice (represented by subscript A) who prepares quantum signals at a transmitter, and Bob (represented by subscript B) who measures the signals at a receiver. A third party, Eve, is considered malicious with the objective to eavesdrop on the quantum channel and/or detector.
In the CV domain, quantum information is usually encoded on the electric field quadratures of light, where the quadrature values X and P represent the in-phase and out-of-phase components of the electric field. The coherent state, which can be labelled | α 〉 = | X + i P 〉 , is usually adopted for the encoding. This state is defined as [43],
| α 〉 = e − 1 2 | α | 2 ∑ n = 0 ∞ α n n ! | n 〉 ,
in the basis of Fock states | n 〉 , where α is the amplitude, and n is the photon number. The encoded quantum signal (henceforth referred to simply as the signal) can be measured using homodyne or heterodyne detection at the receiver. Homodyne detection involves measurement of either the X or P quadrature, while heterodyne detection involves measuring both the X and P quadratures simultaneously. In what is to follow, unless otherwise stated, we will assume homodyne detection is used. Consideration of heterodyne measurements follow a similar path to what is given, with the simultaneous measurement of both quadratures incurring an additional noise penalty [44]. Figure 1(a) illustrates the so-called phase-space diagram for encoding in coherent states, where we have defined the angle θ and amplitude α of the states as θ = arctan ( P / X ) and α = ( X 2 + P 2 ) 1 / 2 . We caution that the development of a "phase operator" in quantum optics has a somewhat complex history and care must be used in interpreting the angle θ in regard to the classical phase ϕ of the electromagnetic wave. A phase operator can be developed which shows that θ → ϕ in the large photon limit (classical limit). When we discuss later the use of phase measurements on reference pulses, we can consider that we are measuring ϕ on those pulses, with ϕ = 0 being defined as the reference pulse phase at the transmitter. For a more detailed description of coherent states, the phase-space representation of quantum states, the development of a phase operator, and the relationship between θ and ϕ , the reader is referred to [43].
CV-QKD protocols can, to a large extent, be separated into two main categories, GMCS CV-QKD and DM CV-QKD. GMCS CV-QKD involves randomly selecting quadrature values from independent Gaussian distributions with modulation variance V m o d , X , P ∈ N ( 0 , V m o d ) , which can be seen in Figure 1(a), where the Gaussian distribution for X is shown in red and the Gaussian distribution for P is shown in blue. Different to GMCS CV-QKD, DM-CV-QKD involves encoding discrete values of the quadratures onto the signals. Encoding is usually achieved by modulating the quadratures through phase-shift keying (PSK) or quadrature amplitude modulation (QAM). In such encoding, a trade-off between increased information transfer and increased difficulty in measuring a larger pool of discretized states will be in effect. As example cases, two of the most common PSK protocols are the four-state quadrature PSK protocol, given in Figure 1(b), and the eight-state 8PSK protocol, given in Figure 1(c). It is important to note that GMCS CV-QKD security proofs are more mature than those of DM CV-QKD [45,46]. However, DM CV-QKD has shown higher reconciliation efficiencies than GMCS CV-QKD at longer distances [47,48] and at lower signal-to-noise ratios (SNRs) [49].
A generic CV-QKD protocol, which applies equally to GMCS and DM CV-QKD, involves the following steps:
  • Alice prepares and transmits states | α 〉 = | X A + i P A 〉 encoded on the signal across a channel (e.g., optical fiber or FSO).
  • Bob measures X B or P B of the signal using his homodyne detector.
  • Key sifting is performed, whereby Alice and Bob decide which variables are to be used for key generation, discarding any uncorrelated measurements.
  • Parameter estimation is undertaken to analyze the system parameters (transmissivity and excess noise), from which it can be determined the amount of mutual information shared by Alice and Bob, as well as how much information Eve has access to.
  • Information reconciliation is carried out, in which, after digitisation of the symbols (using some pre-assigned scheme), an error correction code is used to correct differences in the keys held by Alice and Bob.
  • A confirmation protocol (usually via the use of hash functions) is used to bound the probability that the error correction has failed.
  • Finally, privacy amplification is performed on the keys, shortening their length, to reduce Eve’s information on the key to a pre-assigned negligible level (again, usually via hash functions).
An overview of the generic CV-QKD protocol is given in Figure 2, where the current works on ML-assistance for CV-QKD are highlighted at each step of the protocol.
Note in the above, a pre-shared secret key is used to authenticate all necessary classical communications. The presence of such an a priori shared key is why QKD is perhaps better described as a quantum key growing process. We also note that, for the purpose of the homodyne measurements at Bob, a transmitted local oscillator (TLO) is usually multiplexed with the signal for coherent measurement at the receiver. However, TLOs can be attacked and manipulated by Eve [50]. This has led to the use of real local oscillators (RLOs), also known as local local oscillators, which are phase-corrected using reference pulse information multiplexed with the signals. This latter approach using RLOs can be considered state-of-the-art for CV-QKD [51,52]. We have not attempted to provide a complete methodology of CV-QKD here. A more in-depth explanation of relevant CV-QKD protocols and procedures can be found in the recent survey on CV-QKD [53], while the practical implementation of GMCS CV-QKD is outlined in [44], and a security analysis of DM CV-QKD is given in [46].
Figure 3 illustrates a specific schematic of the preparation, transmission, and measurement of coherent states, including phase compensation using an RLO-based scheme, described as follows. At Alice, a laser source, L A , is passed through a beam splitter to generate the light source (coherent states) for the signals and reference pulses. The (quantum) signals are encoded with randomly selected quadrature values, X A or P A , while the much stronger (classical) reference pulses are encoded with pre-assigned quadrature values (known publicly). The signals and reference pulses are then polarization multiplexed using a beam combiner and transmitted across the channel to Bob who de-multiplexes them. A laser source, L B , at the receiver is split into two RLOs, RLO 1 and RLO 2 . The first of these, RLO 1 , is used to measure the reference pulses via heterodyne detection. These quadrature measurements allow us a first estimate of the phase error Δ ϕ , the phase difference between the transmitted and received reference pulse [52,54,55,56]. This Δ ϕ is then used to phase-correct the second RLO, RLO 2 . Then, by combining the signal and the phase-corrected RLO 2 in a homodyne detector, one of the signal’s quadratures, X B or P B , is measured (the signal and RLO 2 are delayed in this scheme). Note, in the literature this method of phase-adjustment to the RLO 2 if often termed "phase compensation."
The phase compensation technique described above represents but one type of implementation; however, it is important to realise that in essence all implementations have the same aim - the determination of the signal’s quadratures corrected for errors incurred in the preparation, transmission, and measurement of the signal. In the context of CV-QKD, these errors in the signal’s quadratures translate into excess noise [44], a noise beyond vacuum fluctuations that impact key rates negatively [44].
However, the measured Δ ϕ does not exactly predict the phase error between the signal and RLO 1 - a remnant phase error, Δ ϕ ′ , between the signal and the RLO 2 exists. This remnant phase error is due to several real-world limitations of the phase compensation method outlined above. Firstly, both the signals and reference pulses possess 2-dimensional phase wavefronts, where distortion of the wavefronts is not uniformly distributed. The resulting Δ ϕ value is obtained after integrating across the reference pulse wavefront Φ r p ( x , y ) . As such, phase-correction of the RLO 2 wavefront Φ R L O 2 ( x , y ) using Δ ϕ , leads to inconsistencies across the wavefronts: Φ R L O 2 ( x , y ) ≠ Φ r p ( x , y ) , which contribute to Δ ϕ ′ [52,56]. Other contributions to Δ ϕ ′ are introduced by the independent laser sources L A and L B , through such factors as inaccurate clock synchronization and unequal spectral linewidths of the lasers [54,55]. Furthermore, the signals and reference pulses can interact differently with the channel due to their different polarizations, adding to Δ ϕ ′ , as can contributions associated with the signal and RLO 2 propagating through delay lines [54,55].
In the context of this review, we note that ML has been used for Δ ϕ ′ reduction at different stages of the signal quadrature measurement process, including being embedded in the system such that the Δ ϕ ′ estimate is incorporated into phase-correction of the RLO [12], correction of the measured signal quadratures in post-processing using the estimated Δ ϕ ′ [4,5,14], and applying an estimated correction to the wavefront of the RLO [15]. Moreover, ML offers potential improvements in computational efficiency for key sifting, reconciliation, and key rate estimation procedures, improving the feasibility of real-time CV-QKD. Next, we discuss the relevant ML algorithms in detail.

3. Machine Learning Methods

ML is a subclass of artificial intelligence, where we define ML as an algorithm designed to learn patterns from a data set, without being specifically programmed with a set of rules on how to do so.
ML encompasses a very wide variety of algorithms types, which have been be categorized in several ways, sometimes with fuzzy distinctions between categories. This work does not present an in-depth review of ML methodologies; we would recommend [57] for a recent survey on ML algorithms, their applications and future directions. Instead, we outline some key distinctions between ML algorithm types, then briefly introduce some example algorithms to give readers with limited ML experience some perspective on the works presented in Section 4, Section 5, Section 6 and Section 7.
A first distinction between ML algorithms is that they can either be supervised or unsupervised when learning from data. In supervised ML, data is split into training and testing data sets, whereby a model is constructed to map the relationship between known inputs and known outputs using the training data. The test data is then used as input to the trained model to test its output estimation performance. Unsupervised ML models are designed to analyze unlabeled data, discovering patterns in the data without being given example outputs, generally to cluster data, form associations, or to perform dimensionality reduction.
Secondly, the output of a ML algorithm can either be continuous, forming regression algorithms, or discrete, forming classification algorithms. Further, both regression and classification ML algorithms can either incorporate the time-evolution of a system or not when constructing relationships from the data, representing another distinction between ML algorithm types.
Neural networks (NNs) are a popular subclass of ML, themselves encompassing a wide range of algorithms types, which can be used for supervised or unsupervised learning, classification, regression, and time-series analyses. At their core, NNs are comprised of a number of perceptrons (nodes) within a series of layers, where weighted connections are formed between the perceptrons in consecutive layers. Data is fed into an input layer, then a relationship is mapped to an output layer via one or more hidden layers. Deep learning is a subclass of NN, where the depth of the NN represents the number of hidden layers within it. Consensus on the exact number of hidden layers required for an algorithm to be considered deep is fuzzy, though we will define any NN with greater than three hidden layers as a deep learning algorithm.
Figure 4 outlines a taxonomy of the ML algorithms included in this survey on ML-assisted CV-QKD. As can be seen, all of the algorithm types fall under the title of ML, with regression and classification algorithms forming two distinct categories. The distinction between supervised and unsupervised ML algorithms is identified by the unsupervised algorithms written in italics (with supervised algorithms written in normal text). As mentioned, while NN and time-evolution algorithms are only presented within the regression category, their disconnect from classification applies only to the work contained within this survey. NNs can be used for both classification and time-evolution analyses.
A multi-layer perceptron (MLP) is a simple feedforward NN, which takes a vector as input. Perceptrons in each layer of the NN receive inputs x i from the perceptrons i in the previous layer, manipulating the input data using an activation function ψ ( · ) to calculate the output of each perceptron σ j as,
σ j = ψ ( ∑ i = 1 n x i w i j + b j ) ,
using the weight w i j on the i j -th path, and a bias term b j of the relative importance of the activation function. Backpropagation then passes error information back through the layers to update the network weights and biases during training. The error is calculated layer-by-layer, where a type of optimization (often gradient descent) is used to minimize a cost function (often mean squared error), improving estimation results. Different types of activation functions are used for different purposes. For example, the rectified linear unit function is one of the most commonly used activation functions for hidden layers, where ψ ( x ) = 0 if x < 0 , else ψ ( x ) = x if x ≥ 0 . A linear activation function is commonly used in the output layer of NNs designed for regression, such that the output can take any continuous value, while a sigmoid or softmax activation function is commonly used for classification NNs to return discrete outputs. In the surveyed works, MLPs were applied to excess noise filtering in [11], parameter optimization in [24], reconciliation in [27], and key rate estimation in [28].
Figure 5(a) gives an example of a generic fully-connected MLP, where the connections between the input layer, hidden layer, and output layer are shown, as well as an example perceptron in Figure 5(b), indicating how the inputs and weights from the previous layer are transferred into an output via the activation function (following the path outlined in red).
Convolutional neural networks (CNNs) are another common architecture type of NN, which take an n-dimensional array as input (usually two- or three-dimensional), such that spatial relationships between array elements are exploited. CNNs are generally constructed using three layer types: convolutional layers, pooling layers, and fully-connected layers. In convolutional layers, a kernal (filter) is passed over a sub-array of the input data, sifting out a feature map of the relationships between elements. The pooling layers then decrease the size of the convolved feature map to reduce the network’s computational complexity. Often, pooling layers are flattened into a vector to use as input to fully-connected layers, which construct the model output. Fully-connected layers are the same as hidden layers in an MLP, where each perceptron in each layer is connected to every perceptron in the next. Figure 5(c) gives an example CNN architecture, outlining common layer types and the mapping of a two-dimensional array as input to a vector output. Figure 5(d) then gives an example convolution of an input array to a feature map. Surveyed works include CNNs developed for wavefront correction in [15] and Δ ϕ ′ reduction in [12].
Note that the subsequent survey of literature on ML-assistance for CV-QKD includes works using Bayesian inference methods combined with a type of Kalman filter (KF), defined as ML in [4,5,6,7,8,9,10], as well as combined with particle smoothing, defined as ML in [16,17,18], though does not include literature where Bayesian inference alone has assisted CV-QKD.

3.1. Regression

Encoder-decoder NNs are a type of supervised NN commonly used for regression. The encoder maps relationships from input data points to a reduced feature space, then the decoder uses the reduced feature space to reconstruct a prediction of the output space. For example, image-to-image translation can be undertaken using encoder-decoder NNs, such as transforming a picture from a summer scene to a winter scene. Using the example, the common structure between an image in summer and winter would be encoded into the reduced feature space, while the adaptation to winter would be constructed in the decoder. CNNs are often used as encoder-decoder NNs, where the architecture in Figure 2(c) could be used as the encoder, then an inverse architecture could be used as the decoder. Note that encoder-decoder NNs can also be used for classification. Surveyed works used encoder-decoder NNs for wavefront correction [15] and excess noise reduction [13].

3.2. Classification

One of the most common supervised ML classification algorithms is the k-nearest neighbor (KNN) algorithm. Test data points are connected to their k nearest matching neighbors from the labelled training data by measuring their distances, then classified as being in the group containing the most neighbors. Distance-weighted KNN algorithms extend the original KNN methodology by including a weighting to each of the k-nearest neighbors, such that closer neighbors are prioritised. The value of k should be tuned to minimize misclassification. Turbulence strength classification using a KNN algorithm was undertaken in [11], while DM CV-QKD state classification using KNN algorithms was undertaken by [20,21]. Note that no works on ML-assisted CV-QKD utilized a NN for classification.

3.3. Time-evolution

Recurrent NNs represent a subclass of NN designed to store time-series data in a form of memory, such that dependencies can be mapped between the previous data inputs and the next output instance in the series. Essentially, a feedback loop is created, where the network weights and biases of previous time instances are fed into the activation function of the consecutive time instance, forming recurrent layers. A long short-term memory (LSTM) NN is a type of recurrent NN which incorporates a memory cell, which controls the information fed back into the network at consecutive time instances, retaining or discarding certain information to form long-term dependencies. While LSTMs are considered as unsupervised NNs, they have been described as self-supervised given that they learn relationships from the error of previous time instances. The surveyed works [13] and [19] used an LSTM and type of recurrent NN for excess noise filtering, respectively, while [14] used an LSTM for Δ ϕ ′ reduction.

3.4. Unsupervised Learning

Isolation forest algorithms are a type of unsupervised learning ML algorithm used for anomaly detection (a type of classification). Isolation forest algorithms work by forming isolated trees, where the tree randomly selects a dimension from a data set, then randomly splits the data along that dimension. Each of the new subspaces form their own sub tree. This process is repeated until every data point has been isolated as its own leaf node, then the whole process is repeated for multiple isolated trees, which form a forest. The algorithm traverses each tree, assigning a score to each leaf as a function of its depth in the tree, where it is assumed that anomalous data will be isolated along a shorter path length. The data point with the lowest composite score across all trees in the forest is output as the anomalous data point. Removal of abnormal data points for key sifting using an isolation forest algorithm was undertaken by [26].

4. Gaussian Modulated Coherent State CV-QKD

The primary application of ML for GMCS CV-QKD has been recovering the transmitted quadrature measurements X or P at the receiver. This has been achieved using four different approaches. The first was Δ ϕ ′ reduction, where an estimated Δ ϕ ′ was obtained from the reference pulses, then used to phase-correct the RLO for coherent measurement of the signal [12]. The second was correction of the signal quadratures in post-processing using Δ ϕ ′ estimates [4,5,14]. The third was, more broadly, excess noise filtering of the signal quadratures in post-processing [11,13], not focusing specifically on the phase noise. The fourth was correction of the RLO wavefront using reference pulse wavefront estimations [15]. Note that these approaches to signal recovery for GMCS CV-QKD used regression-type ML algorithms.
A Bayesian inference-unscented KF Δ ϕ ′ reduction technique was developed in [4]. A state space model of reference pulse evolution over time was implemented according to discrete Markovian dynamics, with an included Δ ϕ ′ term. The Bayesian inference-unscented KF model was used to evaluate a probability distribution of Δ ϕ at each time step, using the current and previous heterodyne measurements and Δ ϕ estimations, such that the distribution represented the Δ ϕ ′ distribution, and the mean represented optimal estimation of the Δ ϕ . This process allowed the algorithm to update itself based on information from the reference pulses in real-time. The Bayesian inference-unscented KF method was compared to a standard reference method and an extended KF, where it was found to outperform both of them in terms of excess noise filtering (by reducing the phase noise term), particularly for low SNRs, which resulted in higher key rates. An extension of [4] was published in the work [5], where a joint polarization and Δ ϕ ′ reduction model was developed based on the same Bayesian inference-unscented KF approach. Higher key rates were achieved using the proposed ML algorithm than a presented traditional constant modulus algorithm. The phase correction methodology of [4] was then applied to several works, including [6] for their work on CV-QKD across a 60km optical fiber channel, then again across a 100km optical fiber channel [7], as well as in another work on modulation leakage-free CV-QKD [8]. Further, the work [9] implemented the same methodology in their research on a practical CV-QKD approach with composable keys, and again in another work on modulator vulnerabilities in CV-QKD [10].
Another approach to Δ ϕ ′ reduction was developed in [12] using a CNN architecture. The input to the CNN was discussed as being the reference pulse quadratures, while the output was an estimate of Δ ϕ ′ ; however, the structure of the input was defined as being an array of dimension 5×2, without explicitly defining its contents. Of note, the authors incorporated frequency offsets and phase jitter of L A and L B into their model of Δ ϕ ′ . Results from the CNN were compared with those attained using a KF, showing that the CNN estimated Δ ϕ ′ more accurately than the KF across a range of SNRs.
In order to compensate for signal distortion across an optical fiber channel, the work [14] developed an LSTM NN to learn the long-term dependencies of Δ ϕ ′ over times, predicting Δ θ ′ at current and future time steps. Unlike previous approaches, a TLO was used, where Bob measures the signal quadratures, then trains the NN using the quadrature measurements and historic Δ θ ′ information. The Δ θ ′ estimations were then sent to Alice such that she could reconstruct her quadrature values. It was stated that the advantage of their proposed signal recovery methodology was maintaining security of the CV-QKD system, without requiring additional hardware or quantum resources. Key rates were calculated for the case of perfect Δ θ ′ estimation, no Δ θ ′ estimation, and using the LSTM Δ θ ′ estimation method, where results showed the LSTM-based case approached the perfect Δ θ ′ estimation case across a range of channel lengths, while the case of no Δ θ ′ estimation performed significantly worse.
A noise filtering methodology was developed in [13] using an LSTM and autoencoder to estimate corrected quadrature measurements, where the primary emphasis was on noise introduced by the nonlinear imperfections of the balanced homodyne detector and consequent analogue-to-digital converter. When received signal pulses repetition rates were lower than the homodyne detector’s, the amplitude of the signal was maintained; however, as repetition rates increased, lag of sequential signal pulses overlapped. Further, as the sampling frequency of the analogue-to-digital converter should match that of the signal pulse duration, mismatch of the system bandwidth and sampling frequency led to nonlinear distortions of the quadrature measurements. Increased repetition rates greatly improved the secure key bit rate. The autoencoder was used as a noise filtering NN, where its encoder and decoder architectures were constructed using LSTM layers, such that the LSTM memory component could be used to harness the time-based relationships between signal pulses. The NN was trained on three different signal pulse repetition rates, 250MHz, 500MHz, and 1GHz. While positive key rates were possible without the noise filtering algorithm for the 250MHz case, the 500MHz and 1GHz cases resulted in null key rates. However, once the noise filtering NN algorithm was applied, excess noise was reduced below the null key rate threshold. The results indicated how ML can assist in increasing the secure key bit rate across a channel by compensating for noise introduced by CV-QKD hardware.
An equalisation method, assisted by an MLP, was developed in the work [11] to reduce excess noise in the signal quadrature measurements across an optical fiber channel. The authors then applied their equalisation methodology for an FSO channel, such that a KNN algorithm was used to classify turbulence strength, then a different equalisation MLP was trained for each turbulence class, where each NN had a different set of connection weights and perceptron biases. Results showed that signal quadrature measurement corrections were improved when the classification scheme was implemented, particularly for channels with medium-to-strong turbulence.
An alternative approach to phase error reduction after signal distortion across a satellite-to-ground channel was undertaken in [15]. An encoder-decoder CNN architecture was used to estimate wavefront corrections to apply to an RLO, using only the reference pulse spatial intensity distributions as input to the CNN, in order to achieve coherent measurement of signals. Results showed the estimated corrections were able to increase the coherent efficiencies of the measurements for varying satellite altitudes and turbulence strengths, resulting in non-zero key rates, and removing the need to measure the wavefront directly.

5. Discretely Modulated CV-QKD

Similarly to GMCS CV-QKD, ML algorithms designed for excess noise filtering [19] and Δ ϕ ′ reduction [16] have been applied to DM CV-QKD, where the recovered signal was then classified as one of the discrete states used in the DM protocol (i.e., PSK or QAM states). However, unlike GMCS CV-QKD, classification ML algorithms have also been applied to state discrimination in DM CV-QKD, such that the output of the algorithms is the discrete state classification itself [20,21], as well as modulation format identification [22].
A distance-weighted KNN was used to classify states measured by Bob for a four-state quadrature PSK DM-CV-QKD protocol (as shown in Figure 1(b)) in the work [20]. Errors rates were compared against SNRs for the distance-weighted KNN approach and a traditional maximum likelihood classification scheme, where the KNN classification resulted in lower error rates (misclassifications) across the entire SNR range, with greater improvements at higher SNRs. Secure key rates were also calculated for varying channel lengths and modulation variances, with the ML-assisted classification model outperforming the traditional approach in each case. The improvement in secure key rates using the distance-weighted KNN was more pronounced for longer channels and greater modulation variance values.
A multi-label classification algorithm, based on KNN, was developed in [21] to classify 8PSK states. Each quadrant was given a label L i ( i = 1 , 2 , 3 , 4 ) , where each discrete state uses a single label in quadrant PSK, as depicted in Figure 1(b). However, for 8PSK, states, | α 1 〉 , | α 3 〉 , | α 5 〉 , and | α 7 〉 simultaneously belong to two labels each, as shown in Figure 1(c). A feature vector of Euclidean distances to each of the 8PSK state quadrature locations was constructed for each received state to use as input to the classification algorithm. KNN was performed, then posteriori probabilities were calculated for the condition of the received state belonging to a label, applying the label when a threshold probability was reached. Key rates calculated for a range of channel lengths and modulation variances were compared for the proposed 8PSK classification algorithm, as well as quadrature PSK, 8PSK and GMCS CV-QKD, all without the classification algorithm. It was found that the proposed approach achieved higher key rates than each of the protocols without the classification algorithm, except for across short channels (approximately < 10 km).
One work [16] investigated a DM CV-QKD protocol without sending an RP or TLO, only using an RLO and a noise filtering particle smoother model based on Bayesian inference, which filtered the signal quadrature measurements for Δ θ + Δ θ ′ reduction. The Bayesian particle smoother was trained using a Monte Carlo Markov chain method, which optimized tracking of dynamic signal distortion using the set of keys shared between Alice and Bob for parameter estimation. Experimental and simulated data were used to test their noise filtering methodology for both binary PSK and quadrature PSK, with the cases of 0%, 5% and 100% of the keys being used to train the Bayesian model. Using simulated data, excess noise was reduced by approximately an order of magnitude at low SNRs when 100 % of keys were shared, compared to 5 % , before converging on similar noise levels at higher SNRs. Results were volatile for 0 % shared measurements at low SNRs. Experimental results showed that for low SNRs, the cases of 5% and 100% of shared measurements had similar noise filtering efficiencies, both improving upon the 0% case. The absence of a TLO or RPs in the protocol presented in [16] could help to mitigate security vulnerabilities introduced by their transmission, as well as reducing system complexity. Two works which used the same methodology were also published [17,18].
While [16,17,18,20,21] applied ML to aid in state discrimination for fixed PSK state modulation schemes, the work [22] investigated a variable modulation scheme for increased network adaptability and receiver flexibility. A density-based spatial clustering of applications with noise (DBSCAN) algorithm was implemented for modulation format identification, which could identify M-QAM protocols, assuming that Alice had not informed Bob of which modulation scheme was being used. DBSCAN classified the received states as being modulated using a 4-QAM, 16-QAM, 64-QAM or 256-QAM scheme, where it was shown that higher M-QAM values resulted in higher key rates across a range of channel lengths. When comparing SNRs with success rates of correct modulation format identification, lower M-QAM values could achieve higher success rates at lower SNRs. For example 100% success rates were attained for 4-QAM at an SNR of ∼ 10 dB, while 256-QAM required an SNR of ∼ 23 dB. The accuracy of DBSCAN was found to increase as the number of training samples increased. One advantage of DBSCAN over other clustering techniques (such as KNN) is that the number of clusters were determined by the algorithm itself, rather than being defined by the user.
Lu et al. [19] implemented a recurrent quantum NN (RQNN) to assist in coherent state ψ ( x , t ) recovery for a two-state coherent polarization CV-QKD protocol, where the input to the RQNN was the signal quadrature measurements and the output was the filtered signal. It was assumed that the probability density function (PDF) of the Schrödinger equation, calculated as,
i ℏ ∂ ψ ( x , t ) ∂ t = − ℏ 2 2 m ∇ 2 ψ ( x , t ) + V ( x , t ) ψ ( x , t ) ,
could act as a stochastic filter of noisy signals, where ℏ is the Planck constant, m is mass and ( x , t ) represent spatial and temporal coordinates of the quantum state. The spatial potential V ( x , t ) of the Schrödinger equation, given by,
V j ( x , t ) = ϕ j ( ν ( t ) ) ∑ i = 1 n w i j ( x , t ) ,
was implemented as the activation function of each perceptron. The recurrent input of the RQNN was an expected error term ν ( t ) , which was used to construct V ( x , t ) via a Gaussian kernel function ϕ ( ν ( t ) ) , where the connection weights between perceptrons were modified to update V j ( x , t ) for each perceptron j. The summed V ( x , t ) from the output of the perceptrons in the second layer was used to update the Schrödinger equation, which was then used to calculate the PDF as | ψ ( x , t ) | 2 . Noise parameters were calibrated at each time step for each noise source, constructing ϕ ( ν ( t ) ) , which was stated as the core process of the RQNN. By adjusting the PDF, the RQNN decreases the expectation error, improving the accuracy of the filtered signal. The noise filtering performance of the RQNN was compared with a KF and an MLP. Similar noise filtering performance was found for all approaches, given unlimited computational time, however, the RQNN was able to achieve the same performance in far less time. Further, when compared with the MLP, the RQNN is an unsupervised ML algorithm, thus does not need to be trained on existing data beforehand and is able to adapt to unknown fluctuations in the channel.

6. Parameter Estimation and Optimization

Beyond the signal recovery and state discrimination methods given in Section 4 and Section 5, effective parameter estimation and optimization can further improve secure key rates for CV-QKD, where ML has been used to assist in predicting the time-evolution of intensity distributions [23], modulation optimization [24], and key estimation for parameter estimation [58].
A support vector regression (SVR) model was adopted in [23], to predict the time-evolution of the intensity of the laser source at the transmitter and TLO at the receiver in their CV-QKD setup. Experimental intensity distributions were used to train and test the SVR model, with a radial basis function kernel used to fit the hyperparameters of the predictive function. The distorted TLO intensity predictions were then used to amplify or attenuate the TLO at the receiver to compensate for the predicted intensity fluctuations. Secure key rates were analyzed across a range of channel lengths for the proposed SVR-based dynamic TLO intensity modulation approach and a traditional stabilization technique. The SVR-based approach achieved higher key rates across all channel lengths, as well as increasing the maximum channel length where non-zero key rates are attained (up to approximately 50km, compared to 45km using the traditional method). The fluctuations in TLO intensity can be exploited by Eve to conceal attacks on the system, highlighting the need for accurate TLO intensity predictions to identify whether signal distortions are channel or hardware-based versus introduced by Eve.
A signal modulation technique in [24] utilized an MLP to optimize modulation variance for a DM-CV-QKD protocol as a function of FSO channel conditions. Specifically, the refractive index structure parameter C n 2 , transmissivity, and channel length were used as input to an MLP to map a relationship to the optimal modulation variance output for the varying channels. A horizontal FSO channel (fixed C n 2 ) was simulated, of length 0 to 11km, with beam wandering, broadening and deformation all accounted for in the calculation of transmissivity. Higher key rates were achieved when the MLP-based signal modulation optimization was applied, compared to a traditional local search algorithm, though it was not stated which local search algorithm was tested.
The work [58] used an MLP to estimate Alice’s key string by mapping the relationship between a proportion of Alice and Bob’s normalized keys during training, then using the trained MLP to estimate the rest of the key. The objective was to use the estimated key for parameter estimation. Their results indicated that higher key rates could be achieved using their methodology than a claimed conventional scheme.

7. Key Sifting, Reconciliation, and Key Rate Estimation

ML algorithms have also been applied to key sifting [26], information reconciliation [27], and key rate estimation [28,29], with [27,28,29] providing examples of improved computational efficiency of NNs over traditional approaches, while [26] gives an example of ML used for anomaly detection.
One method for key sifting in the work [26] used an isolation forest ML algorithm to sift out abnormal bits, affected by noise, from the raw key. After uncorrelated keys were discarded, the remaining bits were divided into equal-sized miniblocks to input to the isolation forest algorithm, which recursively split the data until each bit was isolated. An anomaly score was calculated as a function of each bit’s height in the miniblock tree to calculate the probability of them being anomalous. Anomalies were introduced to 0.3% of miniblocks of size 10 3 bits, where noise was introduced to 20% of the anomalous miniblocks. Excess noise filtering was compared between the isolation forest method and a traditional Wiener filter for anomalous bit detection, where noise filtering was found to be similar for both methods across a range of channel distances up to 100km. Key rates showed a similar trend, though the Wiener filter approach resulted in slightly higher key rates for most channel conditions. This work presented an example of a traditional algorithm outperforming ML in terms of anomaly detection, highlighting the need to not blindly apply ML algorithms to all problems by assuming that they will achieve the highest performance.
The work [27] developed an MLP to decode a multi-dimensional reconciliation scheme using a low-density parity check code. A Tanner graph was encoded into the MLP, representing the low-density parity check matrix, which was iteratively updated using log-likelihood ratio belief propagation for all connections between the check nodes and variable nodes, then decided whether a bit should have represented 1 or 0. The objective of utilizing the MLP was to optimize the structure of the check matrix, the maximum number of iterations, and the information updating algorithm. The MLP structure was then modified into two alternative decoding algorithms, a linear fitting algorithm optimized to reduce complexity, and a deep NN-assisted decoder optimized to enhance error correction performance. The frame error rate of the original MLP was compared to the linear fitting decoding algorithm and the deep NN decoder using four check matrices, where the deep NN generally outperformed both the original MLP and linear fitting algorithms at error correction, particularly at higher SNRs. A complexity analysis was also performed to highlight the reduced complexity of the linear fitting algorithm, indicating how ML can improve reconciliation speeds for CV-QKD.
Calculating secure key rates traditionally involves numerically minimizing a convex function over all possible eavesdropping attacks. The work [28] developed an ML-assisted key rate estimation methodology for the purpose of reducing computational time and power requirements, improving the real-time practicality of CV-QKD. A minimisation function, designed to optimize the bipartite density matrix shared between Alice and Bob, was encoded as the loss function of an MLP, where the objective of the loss function was to maximize key rate security. The MLP was applied to a quadrature PSK DM-CV-QKD protocol, with the input vector containing 29 variables corresponding to density matrix elements, optimization restrictions, and an excess noise term, then a secure key rate was output. The input vectors were generated for a range of different experimental parameters: channel length, signal intensity, excess noise, and state probabilities. It was shown that calculation time was reduced by up to six orders of magnitude when comparing the MLP to the traditional convex function minimization. Key rates were found to be lower for the MLP-based key rate calculation than for the traditional numerically minimized convex function, though secure probabilities of up to 99.2% were reported, being too insecure for practical CV-QKD. However, the authors stated that, given enough training data, the secure probability could approach traditional key rate estimation methods (in much less time). Taking the methodology of [28] a step further, another work [29] used a Bayesian optimization technique, a tree-structured Parzen estimator, to optimize the structure and hyperparameters of the architecture of a key rate estimation MLP. The output key rates had secure probabilities of 99.15% for quadrature PSK using heterodyne detection and 99.59% using homodyne detection. The optimized MLP architecture was able to reduce key rate calculation times by up to eight orders of magnitude, when compared with the traditional convex function minimization technique, showcasing the primary benefit of the included Parzen estimator.

8. Discussion

Overall, the primary objective of assisting CV-QKD with ML was to increase secure key rates, increasing the practical implementation of CV-QKD. In general, the results from the literature presented this survey found that, when compared with their traditional counterparts, ML was better able to compensate for unknown conditions when recovering transmitted signal information, by performing Δ ϕ ′ reduction, excess noise filtering, discriminating states, estimating and optimizing CV-QKD system parameters, and compensating for key errors.
However, ML should not be considered as an optimal solution to all problems, as shown by the example of a traditional algorithm showing improved results over the ML-based approach [26]. Some works also did not compare their ML methodologies with traditional approaches [11,13,14,15,16,21,22,23], instead focusing on improvements compared to the use of alternative ML algorithms or no algorithm. It should also be noted that the most state-of-the-art traditional methods may not have been tested, though equally, the best ML models may also have not been applied.
An overview of the literature on ML-assistance for CV-QKD is given in Table A1 in Appendix A, outlining each work’s objective, the ML algorithm(s) applied, algorithm comparisons, channel type, and assumptions (further discussed below).
Consideration should also be given to the applied assumptions in the presented works analyzing the benefits of ML assistance for CV-QKD, where performance of the ML algorithms could vary as system fidelity is improved. Moreover, the architecture of the ML algorithms should be constructed deliberately, where an approach of bigger is better does not automatically apply.

8.1. Assumptions

Consideration of the time-domain changes the type of ML model used to assist in CV-QKD. While some literature investigated independent instances of the signal propagation and detection [12,15,20,21,22], in reality there would be time-dependent variations in channel and detection parameters, as well as varying signal modulation. Inclusion of the time domain provides additional information to learn from, likely resulting in higher accuracy model predictions. As such, several of the reviewed works implemented ML models which learnt from the time-evolution of the system [4,5,6,7,8,9,10,11,13,14,16,17,18,19,23].
The channel type has a big impact on the noise affecting the signals, where the two main types are atmospheric FSO and optical fibre-based. The majority of literature in this survey involved signal transmission across simulated [12,13,20,21,22,26,27,28,29,58] or experimental [4,5,6,7,8,9,10,11,14,16,17,18,23] optical fibre channels, with simulated FSO channels explored in [15,24] and experimentally in [11,19]. For example, turbulence fluctuations are highly volatile in atmospheric FSO CV-QKD, where ML-based algorithms designed to compensate for noise could prove more adaptable to unknown channel conditions than traditional model-based approaches.
Another commonly applied assumption applied was whether key rates were calculated at the asymptotic limit [4,5,6,7,8,9,10,11,15,16,17,18,21,28,29,58], or if finite [13,14,20,23,26], or composable effects [9] were taken into account, though this factor may not directly influence the choice of ML algorithm used.

8.2. ML Architecture

The selection and design of ML algorithms requires careful consideration of its application, input and output variables, and modeled system complexity. While this applies to all ML algorithms, we will focus on the use of NNs in this discussion, being the most common category of ML algorithm used in the presented works.
When the input of a NN model is a single value or short vector, such as the measured quadrature X or P values, and the output is a single value, such as the filtered X or P value, then a simple NN structure should suffice. For example, works [11,24] only had one hidden layer in their MLP architectures, with a single value output. Models considering the time-evolution of a system should inevitably be more complex, given that they require historic information to be utilized, such as using LSTM layers in [13] as the encoder and decoder, or the two recurrent layers used in [19]. Increasing the channel complexity, such as for atmospheric channels when compared to optical fiber channels, could also warrant increasing the complexity of the NN architecture as the variation in signal distortions is greater and caused by a more complex process. An example of a deep learning model for CV-QKD was the CNN model used in [15], where the wavefront corrections mapped a relationship between an input spatial distribution of array size 256 × 256 and an output spatial distribution of array size 256 × 256 , requiring a deep network structure. Further, the signals were propagated across a vertical atmospheric channel, which varied as a function of altitude, increasing the complexity of the relationship between the input and output distributions.
The architecture of the MLP in [27] offers an example of a NN designed specifically for the complexity of its application, where its structure was designed to map the Tanner graph of the check matrix of an error correction code into the internal MLP connections between perceptrons. Another example of a NN designed for its application was in [19], where the NN architecture was designed to update the Schrödinger equation by encoding the spatial potential as the activation function, which could potentially improve modeling of a quantum system’s evolution. On the contrary, Xing et al. [12] did not explicitly define their input variables for their CNN, other than its array shape of 5×2, making it difficult to replicate their results and understand their choice of ML algorithm.
Given that the feasibility of widespread CV-QKD relies on its real-time implementation, the complexity of the trained NN model should also be considered (as in [27]). Increasing the complexity of the NN increases the computational requirements to output results, reducing the real-time capabilities of the ML-based approaches.
A final note on the ML algorithms utilized for GMCS CV-QKD versus DM CV-QKD, although continuous quadrature values are encoded in GMCS CV-QKD and discrete quadrature values are encoded for DM CV-QKD, distortion of the signals is continuous. For this reason, regression ML algorithms were applied to GMCS CV-QKD, while both classification ML algorithms [20,21,22] and regression ML algorithms [16,17,18,19] were applied to DM CV-QKD. Classification was used for state discrimination, while regression was used in the same manner as GMCS CV-QKD, correcting the distorted signals, before classifying them using traditional techniques. As such, the regression ML algorithms applied to DM CV-QKD could equally be applied to GMCS CV-QKD.

9. Suggested Future Work

Given that ML-assisted CV-QKD remains a burgeoning field of research, there is currently a wide scope for future work.
In terms of ML algorithm types and structures, there has been limited contrasting of different ML algorithms used for the same application thus far, making it difficult to know whether the ML algorithms used in the reviewed works could have been improved upon with alternative state-of-the-art ML algorithms. For example, no classification-based NN models have been used to estimate DM CV-QKD states, which could potentially match or improve upon KNN or Bayesian-based models. Further, works with ML architectures constructed for their exact application remains limited [19,27]. Application of the Schrödinger equation as the activation function in NNs could be further explored to determine if it can more effectively model signal evolution, such as applying the RQNN designed in [19] to Δ ϕ ′ reduction in GMCS CV-QKD.
Although valid methodologies for achieving CV-QKD can be defined without analyzing the time-evolution of the signal or channel and hardware parameters, which would be ideal for situations where system parameter variation is time-independent, in reality there would be some time-dependence. Work should be undertaken to assess the advantages of applying ML algorithms, which incorporate time-evolution, when compared to time-independent algorithms.
The potential for reference pulse and TLO-less phase compensation and Δ ϕ ′ reduction (as demonstrated in [16,17,18]) should be expanded to better understand its feasibility, while also validating the previous works. Successful implementation of a Δ θ compensation system, without the need for a TLO or reference pulses, could improve CV-QKD security and help to accelerate its practical application.
In general, there are few works on ML-assisted CV-QKD for FSO channels [11,15,19,24], even fewer with experimental results [11,19], and only one satellite-based work [15]. In order to implement a global QKD network, FSO-based CV-QKD, incorporating satellite-based infrastructure, could prove to be the most efficient approach. As such, further investigation of ML to assist in signal recovery and parameter estimation across complex atmospheric FSO channels should be undertaken to better understand its feasibility and limitations.
Finally, given that one of the speculated advantages of ML algorithms is the reduction of computational time and power, when compared to their traditional counterparts, more work should be done to analyze the computational requirements of the ML algorithms, as well as linking those requirements to the practical requirements of CV-QKD hardware.

10. Conclusions

A comprehensive survey of ML algorithms designed to assist in CV-QKD was undertaken, covering a range of applications, including phase error estimation, excess noise estimation, state discrimination, parameter estimation and optimization, key sifting, information reconciliation, and key rate estimation. Results from the literature generally indicated that ML algorithms were able to outperform their traditional counterparts, with respect to noise filtering, parameter optimization, and system estimation, though instances of traditional algorithms outperforming or equaling the performance of ML were also found. An analysis of ML algorithm applicability to different protocols, channel types, and assumptions was undertaken, as well as suggesting future research directions. Overall, from the current field of literature, it can be concluded that ML has been shown to be a powerful tool in assisting to make real-time CV-QKD a practical reality.

Author Contributions

Conceptualization, N.L., R.M., K.G.; methodology, N.L., R.M.; investigation, N.L.; resources, N.L.; data curation, N.L.; writing—original draft preparation, N.L.; writing—review and editing, N.L. R.M, K.G; visualization, N.L.; supervision, R.M, K.G; project administration, R.M.; funding acquisition, R.M., K.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Defence Science and Technology Group, Next Generation Technology Fund, grant number MyIP 8672.

Data Availability Statement

Available on reasonable request.

Conflicts of Interest

The authors declare no conflict of interest. The funding agency had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Abbreviations

The following abbreviations are used in this manuscript:
CNN Convolutional neural network
CV-QKD Continuous-variable quantum key distribution
DBSCAN Density-based spatial clustering of applications with noise
DM Discretely-modulated
FSO Free-space optical
GMCS Gaussian modulated coherent state
KF Kalman filter
KNN K-nearest neighbor
LSTM Long short-term memory networks
ML Machine learning
MLP Multi-layer perceptron
NN Neural network
PDF Probability density function
PSK Phase-shift keying
QAM Quadrature amplitude modulation
RLO Real local oscillator
RQNN Recurrent quantum neural network
SVR Support vector regression
TLO Transmitted local oscillator

Appendix A. ML-Assisted CV-QKD Literature Summary

Table A1. Literature summary.
Table A1. Literature summary.
Work Objective ML Algorithm Algorithm Comparisons Channel Type Assumptions
[4] Δ ϕ ′ reduction Bayesian inference + unscented KF Standard reference method, extended KF Optical fiber Asymptotic key rate, time-domain, experimental & simulation
[5] Δ ϕ ′ reduction Bayesian inference + unscented KF Constant modulus algorithm Optical fiber Asymptotic key rate, time-domain, experimental
[6] Δ ϕ ′ reduction Bayesian inference + unscented KF - Optical fiber Asymptotic key rate, time-domain, experimental
[7] Δ ϕ ′ reduction Bayesian inference + unscented KF - Optical fiber Asymptotic key rate, time-domain, experimental
[8] Δ ϕ ′ reduction Bayesian inference + unscented KF - Optical fiber Asymptotic key rate, time-domain, experimental
[9] Δ ϕ ′ reduction Bayesian inference + unscented KF - Optical fiber Asymptotic & composable key rate, time-domain, experimental
[10] Δ ϕ ′ reduction Bayesian inference + unscented KF - Optical fiber Asymptotic key rate, time-domain, experimental
 [12] Δ ϕ ′ reduction CNN KF Optical fiber No time-domain, simulation
 [14] Δ ϕ ′ reduction LSTM - Optical fiber Finite key rate, time-domain, experimental
 [13] Noise filtering LSTM + autoencoder - Optical fiber Finite key rate, time-domain, simulation
 [11] Noise filtering KNN + MLP - Optical fiber, FSO Asymptotic key rate, time-domain, experimental
 [15] Wavefront correction CNN - FSO (satellite-to-ground) Asymptotic key rate, no time-domain, simulation
 [20] State classification Distance-weighted KNN - Optical fiber Finite key rate, no time-domain, simulation
 [21] State classification Multi-label classification algorithm (KNN) - Optical fiber Asymptotic key rate, no time-domain, simulation
 [16] Noise filtering Bayesian inference + particle smoother - Optical fiber Asymptotic key rate, time-domain, experimental & simulation
 [17] Noise filtering Bayesian inference + particle smoother - Optical fiber Asymptotic key rate, time-domain, experimental
 [18] Noise filtering Bayesian inference + particle smoother - Optical fiber Asymptotic key rate, time-domain, experimental
 [22] Modulation format identification DBSCAN KNN, BIRCH, CLARANS Optical fiber No time-domain, simulation
 [19] Noise filtering RQNN KF, MLP FSO Time-domain, experimental
 [23] Parameter estimation SVR - Optical fiber Finite key rate, time-domain, experimental
 [24] Parameter optimization MLP - FSO No time-domain, simulation
 [58] Parameter estimation MLP Conventional scheme Optical fiber Asymptotic key rate, no time-domain
 [26] Key sifting Isolation forest Wiener filter, COPOD, HBOS, LOF, KNN, MCD, ABOD, PCA Optical fiber Finite key rate, no time-domain, simulation
 [27] Reconciliation MLP, deep NN - Optical fiber No time-domain, simulation
 [28] Key rate estimation MLP - Optical fiber Asymptotic key rate, no time-domain, simulation
 [29] Key rate estimation MLP + Parzen estimator - Optical fiber Asymptotic key rate, no time-domain, simulation

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Figure 1. Coherent states encoded in electric field quadratures for: (a) GMCS protocol showing α and θ , (b) DM quadrature PSK protocol, and (c) DM 8PSK protocol. The size of the circles represent the size of the vacuum noise.
Figure 1. Coherent states encoded in electric field quadratures for: (a) GMCS protocol showing α and θ , (b) DM quadrature PSK protocol, and (c) DM 8PSK protocol. The size of the circles represent the size of the vacuum noise.
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Figure 2. Generic CV-QKD protocol outlining the ML-assisted works for each step of the protocol.
Figure 2. Generic CV-QKD protocol outlining the ML-assisted works for each step of the protocol.
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Figure 3. A schematic outlining the preparation and measurement of coherent states with phase compensation using reference pulses. L is laser source, BS is beam splitter, AM is amplitude modulator, PM is phase modulator, OA is optical attenuater, PBC is polarised beam combiner, PBS is polarised beam splitter, Het is heterodyne detector, Hom is homodyne detector, and DL is delay line.
Figure 3. A schematic outlining the preparation and measurement of coherent states with phase compensation using reference pulses. L is laser source, BS is beam splitter, AM is amplitude modulator, PM is phase modulator, OA is optical attenuater, PBC is polarised beam combiner, PBS is polarised beam splitter, Het is heterodyne detector, Hom is homodyne detector, and DL is delay line.
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Figure 4. Taxonomy of ML algorithms used in the works in this survey. Regression and classification -type algorithms form distinct categories, while subclasses of NNs and time-evolution algorithms can fall within either category and overlap. Unsupervised ML algorithms are written in italics. Acronyms are defined in the following sections and in the list of Abbreviations at the end of this work.
Figure 4. Taxonomy of ML algorithms used in the works in this survey. Regression and classification -type algorithms form distinct categories, while subclasses of NNs and time-evolution algorithms can fall within either category and overlap. Unsupervised ML algorithms are written in italics. Acronyms are defined in the following sections and in the list of Abbreviations at the end of this work.
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Figure 5. (a) Fully-connected MLP schematic with (b) example perceptron outlining the inputs x i , weights w i j , biases b j , activation function ψ ( · ) , and outputs σ j , isolating the path marked in red. (c) CNN schematic with (d) example convolution, isolating the kernel operation marked in red.
Figure 5. (a) Fully-connected MLP schematic with (b) example perceptron outlining the inputs x i , weights w i j , biases b j , activation function ψ ( · ) , and outputs σ j , isolating the path marked in red. (c) CNN schematic with (d) example convolution, isolating the kernel operation marked in red.
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