Coherent optical communications enhanced by machine intelligence

Nonorthogonality of coherent states allows for their use in various quantum and classical optical technology systems, such as quantum networks (Loock 2011) and quantum metrology (Wiseman & Milburn 2009, Armen et al 2002), as well as classical optical communications schemes (Grosshans et al 2003, Betti et al 1995, Grosshans & Grangier 2002). Coherent states are relatively tolerant to losses and nonlinearities in communications channels (Giovannetti et al 2004). Additionally, with the introduction of application-specific integrated circuits (ASICS) and field-programmable gate arrays (FPGAs), research interest involving coherent optical communications is rapidly growing (Yoshida et al 2011). Furthermore, coherent optical phase shift keying, in particular QPSK, communications systems have been demonstrated to have high data transfer rates and low complexity (Patnaik & Sahu 2013). Due to such benefits, coherent optical QPSK is extensively employed in terrestrial (Feng et al 2018) as well as inter-satellite communications (Bindushree et al 2014), such as the global positioning system (GPS), and common data links (CDL). However, the accurate discrimination of coherent states is always a barrier for efficient classical communications, in particular where the receiver site detects weak, low signal-to-noise ratio (SNR) coherent states such as in deep space communications (Kaushal & Kaddoum 2017). Realistic coherent optical communications systems involve the propagation of such signals through various turbulent media that may decrease the SNR (Ke et al 2018), thus increasing the difficulty in discriminating between the four possible phase-shifted keys. As a result, the noisy received QPSK signals can significantly limit the ability to establish such communications in a real-world environment. Here we make use of generative machine learning techniques (Vincent et al 2008) in combination with a convolutional neural network (Lohani et al 2018) to design a communications system that is robust to signal fading (or weak coherent signals) and demonstrate its ability to significantly reduce the error probability in discriminating between received coherent QPSK signals in a realistic simulated communications setting.

Generative machine intelligence systems have recently been applied to a variety of research areas (Sanchez-Lengeling & Aspuru-Guzik 2018, Jang et al 2018, Li et al 2018, Donahue et al 2018, Torlai et al 2018). Here we implement an unsupervised denoising autoencoder (Vincent et al 2008) as the generative neural network (GNN), a concept which has been demonstrated to be useful in multiple applications (Gondara 2016, Fichou & Morlock 2018, Cheng et al 2018). Recently, several research groups have shown the power of machine intelligence in the context of coherent optical communications (Wang et al 2017, Zhang et al 2018, Lohani & Glasser 2018, Khan et al 2017, Fan & Wu 2017, Kulin et al 2017). Here we use a generative neural network, in combination with a convolutional neural network (CNN), to make clean reconstructions and accurate classifications of received optical QPSK signals even for extremely low SNR scenarios, which greatly reduces the overall error probability of the communications design, achieving the classical optimal (i.e homodyne limit (Proakis & Salehi 2001)) or standard quantum limit (SQL). In previous optical communications techniques using machine learning, unknown received optical signals (including images) have been classified with high accuracy using only a CNN as the classifier. It is, however, desirable for schemes with only a CNN as the classifier to have a training set with a large number of known homodyne measurements, which is always unbounded. This limits the discrimination efficiency of the communications design with respect to random signal (amplitude) fading of coherent states. Additionally, such setups with only a CNN as a classifier are based on supervised learning that requires a labelling for each training homodyne measurement. In practice, it is hard to accurately label the homodyne outputs of a received QPSK signal, particularly for very low SNRs, which may require an extra detector that further adds some discrimination error increasing the overall error probability of the communication design. The present unsupervised learning strategy does not require any labelling for the weak (low SNR) coherent states, which are autonomously fed to the networks and reconstructed at the GNN outputs. Afterwards, the generated keys are classified by a CNN purely trained with the homodyne data associated with high SNR QPSK signals (desired), which are easily distinguished and labelled, prior to the start of communications (i.e. pre-trained before messages are sent and received). These novel aspects of the current design pave the way towards the robust and realistic implementation of homodyne receivers in efficiently demodulating signals in the coherent free space optical communications regime.

The overall communications system design is shown in figure 1. A laser beam is split into two paths, path I and path II, with a beam splitter. The branches I and II are used in training the networks prior to communication. Here the branching ensures the realistic implementation of an unsupervised learning (autonomous learning) with the neural networks setup. Note that the QPSK signals $|\alpha^\prime\rangle$, and $|\alpha\rangle$ on path I and II, respectively, share the same QPSK state (key value) so that the homodyne detectors on both paths always read the same key value regardless the strength of noise (SNR) in either branch as shown in figure 1(a). The QPSK signal $|\alpha_{m}\rangle$ ('m' for message) on the communications channel is assumed to be an unknown message key that has been sent and received at the receiver end as shown in figure 1(b). In the prior communication setup, the homodyne outputs of the QPSK signals $|\alpha\rangle$ on path II is fed to the input of the GNN, which reconstructs a new signal measurement at its output that is finally compared with the homodyne output of signals $|\alpha^\prime\rangle$ from path I. Next we optimize the reconstruction loss and update the GNN parameter space (see 'Methods'). At the same time we use the homodyne measurements of the QPSK signals $|\alpha^\prime\rangle$ to train the classifying CNN network (see 'Methods'). Note that the networks are pre-trained, and this pre-training may take place locally, then be distributed prior to implementation of the communications (which is the weak signal $|\alpha_{m}\rangle$ sent). Finally, the machine intelligence aided homodyne receiver classifies unknown QPSK message signals $|\alpha_{m}\rangle$ at the receiver and the corresponding error probability is evaluated. Examples of the homodyne measurement of a received QPSK signal and corresponding clean reconstruction are shown in figure 2. Furthermore, we evaluate the error probability for various combinations of $|\alpha\rangle$ (or $|\alpha_m\rangle$) and $|\alpha^\prime\rangle$, various scanning phase ranges of the local oscillator, various SNR values of the transmitted message signal $|\alpha_m\rangle$, and wide range of phase noise. Finally, we show a significant improvement in the overall error probability when the neural network system is used.

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2.1. Architecture of the GNN

The GNN consists of three sections—an encoder, latent space, and a decoder (generator). The encoder begins with a two dimensional convolutional layer with a kernel of size [5,5], stride length of 1, batch size of 10, feature mappings of 20, same padding, and a ReLU activation function. After this we apply a dropout with a rate of 20% followed by a max-pool layer with a kernel of size [2,2] with a stride length of 2. Then, again, we connect a two dimensional convolutional layer to the max-pool with the same parameters and dropout as discussed above. Next we attach a fully connected layer with the ReLU activation function, where a number of neurons is always equal to the size of the latent space. Note that for the consistency we always choose the size of the latent space as half of the input dimension (Lohani & Glasser 2019). For example, input dimensions of 30 × 30, and 16 × 16, respectively, have a 15 × 15, and 8 × 8 sized latent space. Finally, the encoder convolutes the input dimension of [10 (batch size), width, height, 1 (input channel)] to [10, latent space dimension]. At the beginning of the decoder, the latent space is connected to a fully connected layer with [width/2× height/2× 20 (feature mappings)] neurons, which is followed by a dropout with a rate of 20%. After this we apply a convolutional layer with stride length of 1 and transpose-convolutional (or deconvolutional) layer with stride length of 2, each with a kernel of size [5,5], feature mappings of 20, same padding, and a ReLU activation function. Note that each layer is followed by a dropout with a rate of 20%. Finally a convolutional unit with a single feature mapping generates a clean, or less noisy, homodyne measurement as the output. Here, the generator decodes the latent space of size [10 (batch size), latent space dimension] into the outputs of size [10 (batch size), width, height]. In order to optimize the clean reconstructions, a square reconstruction loss (error) L $\Big(GNN(\alpha),GNN(\alpha^\prime)\Big)$ is evaluated, where GNN(α) is the output of the GNN, and $GNN(\alpha^\prime)$ is the target. Finally, we minimize the average reconstruction loss given by equation (1) using adamoptimizer of tensorflow (Abadi et al 2015) with a learning rate of 0.008.

$$\begin{eqnarray} &&\theta\,,\theta^\prime = {\theta\,,\theta^\prime}\textrm{argmin}\, \frac{1}{N}\sum_i^N {\textbf{L}}\Big(GNN(\alpha),GNN(\alpha^\prime)\Big), \end{eqnarray} \tag{ 1 }$$

where θ and $\theta^\prime$ represent the encoder and decoder parameter space, respectively. The schematic of the GNN is shown in figure 2.

2.2. Architecture of the classifying CNN

Here we simulate QPSK coherent signals $|{\alpha ^\prime }^k\rangle$, $|\alpha^k\rangle$ and $|\alpha_m^k\rangle$ for k ∈ {1,2,3,4} with,

$$\begin{array}{l} {\alpha ^\prime }^k = |{\alpha ^\prime }|{e^{i{\phi ^k}}}{\mkern 1mu} ,{\alpha ^k} = |\alpha |{e^{i{\phi ^k}}},\quad {\rm{and}}\quad \alpha _m^k = |{\alpha _m}|{e^{i{\phi ^k}}},\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{such}}{\mkern 1mu} {\rm{that}}\quad {\phi ^k} = (k - \frac{1}{2}){\mkern 1mu} \frac{\pi }{2}, \end{array} \tag{ 2 }$$

where each coherent state has some mean amplitude (e.g. $|\alpha|$) and phase (φ). In order to simulate a balanced (one output from a 50:50 beam splitter is subtracted from the other as shown in figure 1) homodyne measurement, we use a local oscillator, $\beta\, = \,|\beta|e^{i\gamma}$, with amplitude $|\beta| \, >> |\alpha|$. Note that we set β = 100 for all of the simulations presented in this paper. As a result, the mean signal, $\langle n \rangle$, and variance, (Δn)², at the detector are given by

$$\begin{eqnarray} &&\langle n \rangle = 2|\beta|\langle \alpha| X_\gamma|\alpha \rangle \quad \textrm{and} \quad (\Delta n)^2 = |\beta|^2, \nonumber\\ &&\textrm{given that} \quad X_\gamma = \frac{a^\dagger e^{i\gamma}+ ae^{-i\gamma}}{2},\quad \textrm{and}\quad \gamma :\gamma + \pi/2 \end{eqnarray} \tag{ 3 }$$

where $a^\dagger$ and a are the raising and lowering operators, respectively. Since the minimum error probability (P_err ) of discriminating between the QPSK signals (keys) are bounded below classically by the homodyne limit(Proakis & Salehi 2001), and quantum mechanically by the Helstrom limit(Helstrom 1969), we design and demonstrate that the present communication setups achieve the classical minimum error limit even for extremely low SNR scenarios. In order to calculate the SQL ($P^{HD}_{err}$) and the Helstrom limit ($P^{Hel}_{err}$), we use the following relations, which are discussed in detail in (Proakis & Salehi 2001, Becerra et al 2013, Izumi et al 2012),

$$\begin{align} P^{HD}_{err}\, & = \,\hfill{\textbf{erfc}}(\frac{\alpha}{\sqrt{2}})\Big[1-\frac{1}{4}{\textbf{erfc}}(\frac{\alpha}{\sqrt{2}})\Big];\nonumber\\ P^{Hel}_{err}\, & = \,1 - \frac{1}{8}e^{-|\alpha|^2}\Big(\sqrt{\cosh|\alpha|^2 + \cos|\alpha|^2}\nonumber \\ & +\sqrt{\sinh|\alpha|^2 + \sin|\alpha|^2} + \sqrt{\cosh|\alpha|^2 - \cos|\alpha|^2} +\sqrt{\sinh|\alpha|^2 - \sin |\alpha|^2}\Big)^2, \end{align} \tag{ 4 }$$

where erfc(u)$\, = \,\frac{2}{\sqrt{\pi}}\int_{u}^{\infty}e^{-u^2}\,du$, is the complementary of the error function.

Next, we implement a GNN consisting of three sections—an encoder, latent space, and a generator. The encoder learns the important features of the signals and stores them into a smaller dimension, which is a latent space. The latent space is then decoded through the generator which finally reconstructs the desired clean QPSK homodyne outputs. The implemented GNN uses convolutional units as the encoder and transpose-convolutional units as a decoder. The encoder contains two convolutional layers, a single max-pool layer, and a single fully connected layer (FCL) to the latent space, whereas the generator starts with a FCL followed by a convolutional layer, a transpose-convolutional layer, and again a convolutional layer as shown in figure 2. Similarly, in order to classify the reconstructed, as well as uncorrected (noisy) QPSK homodyne outputs, we use a CNN at the end. The CNN consists of a single convolutional layer followed by max-pooling layer, and two FCLs. Finally, the FCL is connected with an output layer where classification decisions are made as shown in figure 2. The architectures of the GNN and CNN are described in detail in the 'Methods' section. The error probability introduced by the network ($P^{network}_{err}$) is further associated with the error probability while labeling the homodyne outputs for the training sets of the CNN classifier ($P^{label}_{err}$), error probability of the pre-trained CNN ($P^{tCNN}_{err}$), and error probability for making predictions of the unknown message signal ($P^{\alpha_m}_{err}$), which is equal to the ratio of the total number of incorrect classifications to the total number of optical QPSK signal received, resulting in the overall error probability (P_err ) in discriminating the received noisy QPSK signals at the receiver given by,

$$\begin{eqnarray} &&P_{err}\, = \, 1 - (1-P^{HD}_{err})(1-P^{network}_{err}) \nonumber\\ &&= \, 1- (1-P^{HD}_{err})(1-P^{label}_{err})(1-P^{tCNN}_{err})(1-P^{\alpha_m}_{err}). \end{eqnarray} \tag{ 5 }$$

With the aid of the unsupervised GNN in the communication design, we are able to train (pre-train) the CNN classifier such that both $P^{label}_{err}$ and $P^{tCNN}_{err}$ are always 0, which is practically possible with very high SNR data set as discussed in the next paragraph. This leads the overall error probability of the communication design to equation (6),

$$\begin{equation} P_{err}\, = \, 1 - (1-P^{HD}_{err})(1-P^{\alpha_m}_{err}). \end{equation} \tag{ 6 }$$

We then evaluate the error probability and plot it as a ${\textbf{log}}_{10}P_{err}$ and a signal strength $|\alpha|$ in decibels, $|\alpha|\,dB\, = \,10{\textbf{log}}_{10}(|\alpha|)$.

First we evaluate the P_err with respect to the amplitude $|\alpha^\prime|$ along the path I, given different SNRs in path II. Here we scan the local oscillator phase from 0 to 2π with a grid consisting of 900 points. We randomly simulate 200 homodyne outputs for each QPSK signal for 17 different values of $|\alpha^\prime|$ from −15 dB to 9.08 dB, and set them as the target patterns at the output of the GNN. Similarly, we do the same for path II and randomly simulate 200 homodyne outputs for each QPSK signal for $|\alpha|$ = −12.0 dB, −10.5 dB, and −9.3 dB, and feed them to the encoder of GNN. Note that before feeding the signals to the GNN, we convert the homodyne outputs (from path I and II) into corresponding 30 × 30 pixels images as shown in figure 2. In order to optimize $|\alpha^\prime|$ at various values of $|\alpha|$, we keep $|\alpha|$ fixed at −12.0 dB, −10.5 dB, and −9.3 dB, and vary $|\alpha^\prime|$ for all of them separately, such that 17 different GNN configurations are trained up to 150 epochs (approximately 8 minutes) separately for each $|\alpha|$, for a total of 51 pre-trained GNNs. Also, we use the same $|\alpha^\prime|$ data set, separately, to pre-train the classifying CNN networks as shown in figure 1(a). Then the pre-trained networks are distributed to the receiver site (Bob).

In order to establish a communication setup as shown in figure 2(b), we randomly simulate 360 homodyne outputs (90 per each QPSK key) for each $|\alpha_m|$ = −12.0 dB, −10.5 dB, and −9.3 dB as the test sets. Note that message data and training data for the pre-communication setup share no similarity as they are randomly simulated at different times. Finally at the receiver, the homodyne measurements of $|\alpha_m|$ are fed to the distributed pre-trained GNN which generates desired homodyne measurement as the output. After that, in order to calculate P_err , the reconstructed images (generated homodyne measurement outputs) from the GNN are forwarded to a pre-trained CNN and the corresponding P_err is measured, the results of which are shown in figure 3(a). Note that the distributed pre-trained networks reconstruct and then classify $|\alpha_m|$ on the order of milliseconds. The P_err for $|\alpha_m|$ = −12.0 dB, −10.5 dB and −9.3 dB at various $|\alpha^\prime|$ are shown by the solid red, green, and blue curves, respectively, with the same color dotted lines representing the corresponding homodyne limit ($P^{HD}_{err}$), and black solid line representing the Helstrom limit ($P^{Hel}_{err}$). We find an improvement in P_err begins when $|\alpha^\prime|$ ≥ −4.5 dB for all $|\alpha_m|$ = −12.0 dB, −10.5 dB, and −9.3 dB. As expected we obtain better reconstructions and less P_err as we increase $|\alpha^\prime|$ up to −1.5 dB for $|\alpha_m|$ = −12.0 dB, and around 0 dB for, −10.5 dB, and −9.3 dB, after which they begin to saturate. Additionally, we show that the system's P_err achieves the corresponding homodyne limit for $|\alpha_m|$ = −9.3 dB at $|\alpha^\prime|$ = −1.5 dB. Similarly we find a difference of 2.4 × 10⁻³, and 6.8 × 10⁻³, respectively, between the P_err and corresponding homodyne limit for $|\alpha_m|$ = −10.5 dB, and −12.5 dB when $|\alpha^\prime|$ = 0.05 dB, and 6.1 dB. Moreover, the corresponding P_err for $|\alpha^\prime|$ from −2 dB to 4 dB at the given $|\alpha_m|$ are zoomed in and shown in the inset of figure 3(a). After taking into account the various SNR scenarios, we set $|\alpha^\prime|$ relatively high at 9 dB as the optimized value for all simulations and results discussed in the following paragraphs. Note that this signal is used in pre-training the neural networks prior to communication, and is not the signal that would be sent through a realistic communications channel (which is $|\alpha_{m}|$, and is simulated to be much weaker as discussed in the following).

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Next, we investigate the P_err as the phase scanning range of the detection system is varied. We use the same training-test data and network settings as discussed above. In order to simulate training and test sets at various phase scanning ranges, we slice the grid of the homodyne outputs into a number of segments, for example as discussed in previous paragraph, into a grid of 30 × 30 (i.e. 0 − 900 points) which represents a phase scanning from 0 to 2π. Here, we slice it into 28 × 28 (0 − 784 points), 26 × 26 (0 − 676 points), 24 × 24 (0 − 576 points) and so on up to 4 × 4 (0 − 16 points), for a total of 14 various scanning ranges. These correspond to phase scanning ranges from 0 − 1.74π, 0 − 1.5π, 0 − 1.28π, and so on up to 0 − 0.04π, respectively. Note that we separately train the GNN and CNN for each scanning range. Also we set the latent space dimension of the GNN to half of the input dimension for a given range. For example, the input of 28 × 28 and 12 × 12 points, respectively, have latent space sizes of 14 × 14 and 6 × 6 points. With the GNN pre-trained for an $|\alpha|$ = -10.5 dB, and -9.3 dB, and the CNN pre-trained with $|\alpha^\prime|$ = 9 dB, we simulate sending a message signal $|\alpha_m|$ = −10.5 dB, and −9.3 dB (different from training sets) and calculate the P_err at the given scanning range. Here we reconstruct the 360 homodyne measurements (90 per each QPSK) for the given message set $|\alpha_m|$ and predict the corresponding QPSK value using the CNN, which is finally used to evaluate the P_err . The P_err resulting from the networks with and without the GNN versus scanning range is shown in figure 3(b). The improvement in P_err for $|\alpha_m|$ = −10.5 dB is shown by red curves, where the dotted red curve, solid red curve, and horizontal red line represent the P_err without the GNN, with the GNN, and the homodyne limit, respectively. Similarly, the dotted green curve, solid green curve and a horizontal green line represent the P_err without the GNN, with the GNN, and the homodyne limit for $|\alpha_m|$ = −9.3 dB, respectively. As discussed above, the black horizontal line is the Helstrom limit. We find a gradual improvement in the P_err as we increase the scanning phase range of the local oscillator. Furthermore, with the aid of the GNN in the network and a scanning range of 0 − 0.88π, we show significant improvement in P_err from −0.098 to −0.128, and −0.124 to −0.151 for $|\alpha_m|$ = −10.5 dB, and −9.3 dB, respectively. Note that the homodyne limit for $|\alpha_m|$ = −10.5 dB, and −9.3 dB are −0.147 and −0.154, respectively. Additionally, we find the P_err with the aid of the GNN is minimized and achieves the corresponding homodyne limit when the scanning range is greater than or equal to 0 − 1.14π for $|\alpha_m|$ = −9.3 dB. A minimum difference of 2.6 × 10⁻³ between the GNN-aided P_err and homodyne limit is achieved for $|\alpha_m|$ = −10.5 dB at a scanning range of 0 − 2π. The P_err corresponding to a scanning range of 0 − 1.28π, 0 − 1.74π, and 0 − 2π are zoomed in and shown in upper right inset of figure 3(b).

We now turn to investigating the improvement of the P_err at various SNR levels of transmitted message signals $|\alpha_m|$. In order to generate training input sets for the GNN-network, we randomly simulate 200 homodyne measurements for each QPSK key for 15 values of $|\alpha|$ from −15.0 dB to −9.12 dB. As mentioned earlier, we set $|\alpha^\prime|$ to 9 dB, and the corresponding 200 homodyne outputs for each QPSK signal are used as the target for the GNN. The same $|\alpha^\prime|$ is used to train the CNN. Note that we train separately the GNN for each value of $|\alpha|$, giving a total of 15 GNNs, while the classifying network (CNN) remains the same (as they share same target of the GNN). Similarly, we randomly generate 90 homodyne measurements for each QPSK for 15 values of $|\alpha_m|$ from −15.0 dB to −9.12 dB as the message set separately. As a result, message sets and training sets again share no similarity. With the pre-trained GNN and CNN (distributed), we evaluate the P_err with and without the GNN in the networks, results of which are shown in figure 4(a). The dotted red and solid blue curves represent the P_err without the GNN and with the GNN in the networks, respectively. Similarly, a solid green and a black line represent the homodyne and Helstrom limits for various values of $|\alpha_m|$, where as a solid magenta line represents the P_err for any arbitrary classifier or device with an efficiency (η) of 95%. With the aid of the GNN in the networks, we achieve a remarkable improvement in P_err from −0.119 to −0.147, which nearly reaches the optimal homodyne limit of -0.148 for $|\alpha_m|$ = −10.23 dB. Similarly, for $|\alpha_m|$ = −9.27 dB, the P_err achieves an improvement from −0.129 to the optimal homodyne limit as shown in upper right inset of figure 4(a). Furthermore, even for a very low SNR of $|\alpha_m|$ = −13.49 dB, we significantly reduce the difference between P_err and the homodyne limit from 6.9 × 10⁻² to 2.7 × 10⁻². Additionally, for $|\alpha_m|$ ≈ −12 dB, we find that the P_err curve with the GNN in the networks crosses and beats the P_err curve with any 95% efficient detector at the receiver.

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Finally, we significantly lower the P_err for the various strengths of phase noise associated with the transmitted message signals $|\alpha_m|$ and the local oscillator, which is shown in figure 4(b). In order to simulate the phase noise at the receiver, we introduce a phase offset of δ in the mean signal, which is $\langle n \rangle\, = \,2|\alpha||\beta|\textrm{cos}(\phi-\gamma+\delta)$ such that δ ∈ N(0, ρ), a normal distribution with mean 0 and variance ρ². In order to train the networks we keep the signal strength of $|\alpha^\prime|\, = \, 9 dB$ and $|\alpha|\, = \,-9.3 dB$ fixed along the path I and II, respectively, as discussed earlier. As homodyne measurements along the path I are used as the target and training sets for the GNN and classifying CNN, respectively, we assume that this path has only phase noise strength (ρ) of π/36 radians (5°), and randomly simulate 200 homodyne outputs for each QPSK signal. At the same time, we simulate to vary ρ from π/36 to 8π/45 radians (32°) along the path II, and again randomly generate 200 homodyne outputs for each QPSK signal and ρ, which are fed to the input of GNN. Examples of homodyne measurements of the optical QPSK signal along the path I with ρ of 5° (top-left) and 32° (top-right) are shown in the inset of figure 4(b). After the networks are pre-trained and distributed, we randomly simulate 90 homodyne outputs for each QPSK message signal with $|\alpha_m|\, = \,-9.3 dB$ and given ρ. Note that the training homodyne sets of the networks and the message homodyne sets again share no similarity. An example for a message QPSK homodyne output with the ρ of 32° is shown in the bottom-right inset of figure 4(b). First, we train the networks separately for each ρ and evaluate the P_err , the results of which are shown in figure 4(b). The green, black and magenta horizontal lines, respectively, represent the Homodyne limit, Helstrom limit, and P_err introduced by an arbitrary η = 95% efficient detecting device. The blue solid and red dotted curves show the P_err with the GNN and without the GNN in the networks, respectively. With the aid of GNN, we find a significant reduction in the P_err for the wide range of phase noise strengths from 5° to 32°. For example with the addition of GNN, the P_err for the phase noise strength of 8°, 20°, and 26° are reduced to the Homodyne limit.

In conclusion, we have designed a novel coherent optical communications setup that efficiently demodulates weak coherent optical QPSK states with a robust machine intelligence aided balanced homodyne receiver. The developed state-of-the-art GNN and CNN system, in combination, corrects for coherent QPSK signals associated with a wide range of SNRs, resulting in significantly reduced overall error probability of the communications system, either achieving or approaching the classical optimal limit. Additionally, by using the same network system, we also reduce the discrimination error probability for various scanning ranges of the local oscillator at different SNRs. In addition, we illustrate that the network design is robust for the wide range of phase noise associated with the optical signal and the local oscillator as well. Furthermore, with the aid of the unsupervised GNN, which reconstructs a clean and desired quadrature measurement at its output, we have minimized the error probability with a CNN that is exclusively trained with homodyne measurements associated with high SNR (desired) coherent QPSK signals. This allows bypassing the need for extremely large CNN training sets that would require various noises which are not only difficult to label, but are also unbounded. The present advances in QPSK demodulation and classification are essential to the robust performance of realistic coherent optical communications systems, and we anticipate that the presented techniques may directly be applied to an enhancement of current classical and quantum communications protocols (Bacco et al 2019, Eriksson et al 2019).

This material is based upon research supported by, or in part by, the U. S. Office of Naval Research under award number N00014-19-1-2374. Additionally, the research was supported in part using high performance computing (HPC) resources and services provided by Technology Services at Tulane University, New Orleans, LA.

S L conceived and designed the neural network system, and ran all simulations. R T G supervised the project. Both authors prepared the manuscript.

The data that support the findings of this study are available from the corresponding authors on reasonable request.

The authors declare no competing interests.