WO2025048685A1

WO2025048685A1 - Joint training of hybrid quantum-classical autoencoder for csi compression

Info

Publication number: WO2025048685A1
Application number: PCT/SE2023/050857
Authority: WO
Inventors: Ahsan Javed AWAN; Sairamesh Nammi
Original assignee: Telefonaktiebolaget LM Ericsson AB
Current assignee: Telefonaktiebolaget LM Ericsson AB
Priority date: 2023-08-25
Filing date: 2023-08-25
Publication date: 2025-03-06
Anticipated expiration: 2026-02-25

Abstract

A method (100), performed by a network node, for enabling CSI compression. The method comprises clustering (101) UEs according to a measure of similarity, training (102) a hybrid autoencoder comprising a classical encoder and a quantum decoder, for each cluster of UEs, on training data corresponding to CSI of a selected UE from the cluster of UEs, and transmitting (103) the encoder of the hybrid quantum-classical autoencoder to each UE in the cluster of UEs. Also disclosed are related network nodes, radio access nodes, computer programs, and computer program products.

Description

JOINT TRAINING OF HYBRID AUTOENCODER FOR CSI COMPRESSION TECHNICAL FIELD The disclosure relates generally to methods for compression of channel state information in a telecommunications network and more specifically relates to methods for training and deployment of a hybrid autoencoder for compression of channel state information. Further disclosed are a related network node, computer programs, and computer program products. BACKGROUND In frequency division duplex, FDD, mode of massive multiple-input multiple-output, MIMO, systems, a base station and a user equipment, UE, associated to the base station transmit in the same time slot but at different frequencies. This breaks the reciprocity between uplink channel state information, CSI, and downlink CSI and makes it difficult to obtain an accurate downlink CSI estimate for transmit signal processing. In current 5G New Radio, NR, as defined by the 3^rd Generation Partnership Program, 3GPP, obtaining a downlink CSI estimate is a process initiated by a next- generation Node B, gNB, by transmitting a reference signal specific to the cell and/or UE to the UE. From this reference signal, the UE computes channel estimates and the parameters needed for CSI reporting, such as for example a channel quality indicator, a precoding matrix index, rank information, or a CSI resource indicator. The prepared CSI report is then sent to the network via a feedback channel either on request from the network aperiodically or the UE is configured to send the report periodically. The network scheduler uses this information for choosing scheduling parameters for the UE. The scheduling parameters are transmitted to the UE in a downlink control channel, after which data transmission takes place from the network to the UE. However, transmitting the CSI report to the base station is costly due to bandwidth limitation of the feedback link. The base station has a large number of antennas for massive MIMO, leading to feedback overhead which may be overwhelming for the UE to transmit to the base station. In current implementations of radio networks, this problem is resolved by UE using a codebook based approach, where a pre-defined codebook is known to the base station and the UE. From the codebook, the UE selects the precoding matrix/vector that best matches the estimated CSI from the reference signal. However, the codebook approach only provides the base station with a quantized rough approximation of the actual CSI. For improved network performance, it is therefore of interest to develop methods for the UE to transmit more complete CSI information to the base station, without having to increase the overhead in the uplink direction. One approach to solving this problem is to compress the CSI using a machine learning algorithm. In particular, autoencoder neural networks, ANNs, have been used to solve this problem in prior art. ANN based approaches rely on implementing the encoder and decoder part of a jointly trained autoencoder distributed on the UE and the base station, so that the encoder at the UE maps channel estimates or corresponding observations of pilot data to an appropriate feedback format and the decoder at the base station reconstructs the complete downlink channel estimates based on the received feedback. However, deep learning based approaches such as ANNs are sensitive to the variation of the communication environment and require very large amounts of training data to ensure a reasonable level of performance. Since the training data comprises CSI, obtaining the training data poses the same problem as obtaining the CSI at the base station in the first place. In prior art, one method (Yang, H., Jee, J., Kwon, G. and Park, H., 2020, October. Deep Transfer Learning-Based Adaptive Beamforming for Realistic Communication Channels. In 2020 International Conference on Information and Communication Technology Convergence (ICTC) (pp.1373-1376) IEEE.) to circumvent this problem exploits the information from a pre-trained DNN for training other DNNs to find the beamforming vector from uplink CSI in the specified channel. Another approach (Zeng, J., He, Z., Sun, J., Adebisi, B., Gacanin, H., Gui, G. and Adachi, F., 2021, March. Deep Transfer Learning for 5G Massive MIMO Downlink CSI Feedback. In 2021 IEEE Wireless Communications and Networking Conference (WCNC) (pp.1-5). IEEE.) is to train a neural net using many samples from a model of a non-line-of-sight transmission channel and then fine-tunes the pre-trained model with a small number of actual data points from the channel. Quantum autoencoders can perform machine learning tasks analogous to those performed by classical autoencoders without the exponentially costly use of classical memory. Fig.5 shows an exemplary circuit 500 of a quantum autoencoder. The parameters obtained by training such a circuit are hypothesized to be considerably smaller than the weights obtained by a classical autoencoder, and thus the training load would be smaller. For example, Romero, J., Olson, J.P. and Aspuru-Guzik, A., 2017, Quantum autoencoders for efficient compression of quantum data. Quantum Science and Technology, 2(4), p.045001, demonstrates that a quantum neural network operating on ^ qubits with parameter size ^(log⁽^⁾) can achieve good results with only polylogarithmic training data size. Therefore, simply applying a QNN in a quantum autoencoder could lower training overhead in CSI compression. However, several problems remain with training and deploying a quantum autoencoder in a telecommunications network. Notably, quantum chips in UEs are generally thought to be much further in the future than quantum capabilities in a base station or a quantum computer available to a base station for performing some limited calculations. It may therefore not be preferred to implement a solution reliant on the UE having quantum capabilities. Moreover, training an autoencoder of any type is typically prohibitively expensive for a UE, so training has to be managed centrally. However, obtaining CSI data and training an autoencoder for each UE at the base station would be prohibitively expensive for the base station. It is therefore of interest to develop a method to take advantage of the benefits of QNN-based autoencoders, while taking restrictions in computational power and availability of quantum resources at UE and base station into account. SUMMARY It is an object of the present disclosure to improve CSI reporting. In particular, it is an object of the present disclosure to provide a method for computationally efficient training and distribution of hybrid autoencoders for CSI compression. According to a first aspect of the disclosure, there is a method for enabling compression of channel state information, CSI. The method is performed by a network node of a communication system. The method comprises clustering user equipment, UEs, associated to the network node according to a measure of similarity. The method further comprises training a hybrid autoencoder for each cluster of UEs, the hybrid autoencoder comprising a classical encoder and a quantum decoder, on training data, the training data at least comprising CSI data points corresponding to a selected UE in the cluster of UEs. The method further comprises transmitting each encoder of the trained hybrid autoencoder to each UE of the associated cluster of UEs. Hereby is achieved improved CSI reporting. According to an embodiment of the first aspect, the measure of similarity is based on one or more of: deterministic multipath components of a signal between each UE and the network node; beam indices of a codebook of the network node, wherein the beam indices correspond to beams in a pre-defined beam grid associated to an established channel between each UE and the network node; or a channel covariance matrix of an established channel between each UE and the network node. Hereby is achieved that the encoder is trained on representative training data. According to an embodiment of the first aspect, training the hybrid autoencoder comprises encoding each training data point in the training data set as a quantum state using a state preparation circuit implementing basis encoding. Training the hybrid autoencoder further comprises passing the encoded data point and a reference state representing the target number of qubits desired to compress the encoded data point to an encoder unitary. Training the hybrid autoencoder further comprises training the encoder unitary to maximize the fidelity between trash states of the encoder unitary and the reference state. The embodiment further comprises obtaining a decoder unitary by taking a complex conjugate of the encoder unitary and obtaining the encoder by reducing the trained quantum encoder to a classical feedforward neural network. Hereby is achieved efficient training of the autoencoder. According to an embodiment of the first aspect, the size of the reference state is computed from the dimension of the training data points, a desired compression ratio, and an encoding scheme used in the decoder unitary. Hereby is achieved a desired compression ratio of the autoencoder. According to an embodiment of the first aspect, a loss function used to train both the classical encoder and the quantum decoder is a function of the data encoded in quantum states. Hereby is achieved efficient training of the autoencoder. According to an embodiment of the first aspect, a loss function used to train the classical encoder is a function of the training data before encoding the training data into quantum states and a loss function used to train the quantum decoder is a function of the training data encoded in quantum states. Hereby is achieved higher accuracy of the autoencoder. According to an embodiment of the first aspect, the selected UE for each cluster of UEs is the centroid of the cluster of UEs. Hereby is achieved that representative training data is obtained. According to an embodiment of the first aspect, the method further comprises at predetermined events, finetuning each trained hybrid autoencoder on training data corresponding to a subset of UEs in the corresponding cluster of UEs. Hereby is achieved that the autoencoder is kept accurate over time. According to an embodiment of the first aspect, fine-tuning the hybrid autoencoder comprises finetuning the parameters of the hybrid autoencoder using training data obtained by mixing training data from each of the encoders of the UEs in the cluster except for training data of the selected UE. Hereby is achieved that the finetuning compensates for biases in the selected UE. According to an embodiment of the first aspect, the training data further comprises test batches of data from the encoder of the selected UE. hereby is achieved that the finetuning does not degrade the performance on the selected UE. According to an embodiment of the first aspect, only data from UEs on which the hybrid autoencoder performs below a predetermined threshold is used for finetuning. Hereby is achieved efficient finetuning. According to an embodiment of the first aspect, finetuning is triggered by a change in the moving speed of a subset of the UEs. Hereby is achieved an easy to detect trigger for finetuning. According to an embodiment of the first aspect, the method further comprises obtaining the CSI comprised in the training data by receiving CSI compressed at the UE via random projections and reconstructing the CSI at the network node by solving an L1-norm optimization. Hereby is achieved that training data may be obtained. According to a second aspect of the disclosure, there is a network node of a communication system for enabling compression of channel state information, CSI, the network node comprising a memory and processing circuitry, the processing circuitry comprising a quantum processing unit. The network node is configured to cluster user equipments, UEs, associated to the network node according to a measure of similarity. The network node is further configured to train a hybrid autoencoder for each cluster of UEs, the hybrid autoencoder comprising a classical encoder and a quantum decoder, on training data, the training data at least comprising CSI data points corresponding to a selected UE in the cluster of UEs. The network node is further configured to transmit each encoder of the trained hybrid autoencoder to each UE of the associated cluster of UEs. According to an embodiment of the second aspect, the network node is further configured to perform a method according to any embodiment of the first aspect. According to an embodiment of the second aspect, the network node is a radio access node. According to a third aspect of the disclosure, there is a computer program comprising machine-readable instructions which, when executed on the processor of a network node, cause the network node to perform a method according to any embodiment of the first aspect. According to a fourth aspect of the disclosure, there is a computer program product comprising a computer program according to the third aspect. The advantages of the embodiments of the first aspect apply, mutatis mutandis, to embodiments of the second through fourth aspect of the disclosure. BRIEF DESCRIPTION OF THE DRAWINGS Fig.1 is a flow diagram of a method according to an embodiment of the disclosure. Fig.2 is a handshake diagram of exemplary communication between a network node and a UE according to the disclosure. Fig.3 is a handshake diagram of exemplary communication between a network node and a UE according to an embodiment of the disclosure. Fig.4 is a network node according to embodiments of the disclosure. Fig.5 is an exemplary quantum circuit according to an embodiment of the disclosure. Fig.6 is an exemplary user equipment according to an embodiment of the disclosure. Fig.7 is an exemplary communication system according to an embodiment of the disclosure. DETAILED DESCRIPTION OF THE DRAWINGS Fig.1 depicts a flow diagram of an embodiment of a method 100 according to the disclosure. The method is a computer-implemented method for enabling compression of channel state information, CSI. CSI comprises the known channel properties of a communication link. The channel properties describe how a signal propagates from the transmitter to the receiver. Channel properties may include, for example, the fading distribution of the channel, the average channel gain, the line-of- sight component of the channel, and the spatial correlation between the signal’s spatial direction and the average received signal. Broadly speaking, the CSI characterize an impulse response of the channel. In the context of advanced telecommunications systems such as massive multiple-output multiple-input, MIMO, base stations are equipped with a large number of antenna elements, and precision control of the antenna elements enables functionalities like beamforming, allowing for energy efficiency, better utilization of the available spectrum, and less radiation emissions and hence less interference with other communication systems. A UE informing the base station of the CSI allows the base station to determine how to control the antenna elements to transmit the best beam for the specific UE. Methods according to the disclosure may be performed in any communications network necessitating the transmission of any form of CSI. In particular, the method may be performed in a communication system utilizing some form of MIMO technology, such as wireless local area networking protocols such as Wi-Fi 4, Wi-Fi 5, Wi-Fi 6,and Wi-Fi 7 as defined by the Institute of Electrical and Electronics engineers, or a wireless telecommunications network such as Universal Mobile Telecommunications System, UMTS, Long-Term Evolution, LTE, New Radio, NR, as defined by the 3^rd Generation Partnership Program, 3GPP, or a network according to any other or future standard as defined by a suitable standardizing body. Alternatively or in addition, methods of the disclosure may be implemented in a hybrid network incorporating aspects of several different types of networks. The network implementing methods according to the disclosure may be private networks or public networks. Embodiments of the method 100 are performed by a network node. A network node in a communication network may comprise a physical or virtual unit of the network capable of creating, receiving, and/or transmitting information over a communication channel of the network. A physical network node may for example comprise a modem, a router, or a wireless access point in a Wi-Fi network, or it may comprise a radio access node such as an eNodeB or a gNodeB, or a core network node in a telecommunications system. A virtual network node may some or all the tasks of a physical node, but distributed over a series of client computing devices. In some embodiments of the method, the network node may be an Open RAN, O-RAN node. An O-RAN node is a node that supports an O-RAN specification, such as a specification published by the O-RAN Alliance or a similar organization. The network node may serve a set of UEs. The set of UEs served by the network node may be UEs in a cell or physical area corresponding to the network node. Some UEs of the set of UEs may be served by several network nodes. The UEs may be any of a wide variety of communication devices, including in particular mobile phones, smart phones, cell phones, voice over IP phones, wireless local loop phones, desktop computers, personal digital assistants, wireless cameras, gaming consoles and gaming devices, music storage devices, playback appliances, wearable terminal devices, wireless endpoints, mobile stations, tablets, laptops, laptop-embedded equipment, laptop-mounted equipment, wireless customer- premise equipment, vehicles, vehicle-mounted or vehicle-embedded/integrated devices, narrow band IoT devices, machine type communication devices, and or enhanced machine type communication devices, or internet of things, IoT, devices. In some embodiments, the UEs may be configured to transmit information without direct human interaction. For example, an IoT sensor may be configured to transmit information to a network on a predetermined schedule, or to request messages from a network at a predetermined schedule. The UEs may be configured to operate using a single mode of connectivity, or be configured for multi-radio dual connectivity. A UE may support device-to-device communication in addition to communication with a network node. The method comprises clustering 101 the UEs according to a measure of similarity. Clustering the UEs according to a measure of similarity may comprise clustering the UEs based on deterministic multipath components of a signal between each UE and the network node. Deterministic multipath components of a signal are the components of multipath propagation which may be predicted, such as those caused by large stationary objects like buildings. Alternatively or in addition, clustering the UEs according to a measure of similarity may comprise clustering the UEs based on beam indices of a codebook of the network. In for example LTE and NR telecommunications systems, a codebook comprises a set of precoding matrices which, when applied to a signal at a baseband of a radio access node, form a specific beam. The precoding matrices are indexed, and based on CSI of the channel, the radio access node may select the best precoding matrix for the UE. Clustering the UEs based on beam indices of the codebook thus comprises clustering UEs with similar precoding matrices. Alternatively or in addition, clustering the UEs based on a measure of similarity may comprise clustering the UEs based on a channel covariance matrix of the channel between each UE and the network node. Clustering based on a channel covariance matrix may comprise selecting a reference UE, possibly at random, and computing the eigenvalues of the channel of the reference UE, and cross correlating the eigenvalues with eigenvalues for the other UEs associated with the network node. It further comprises grouping UEs with correlations within a pre-determined threshold. Thresholds for clustering may be set based on available computing resources at the network node. A low threshold may lead to many clusters, but also excellent performance. However, most network nodes do not have unlimited computing resources and implementations may thus select a threshold that leads to a number of clusters the network node has resources to train. The clustering may be performed using any suitable clustering algorithm. For example, centroid models such as k-means or Kernel k-means clustering may be used. Alternatively, density models such as density based spatial clustering of applications with noise, DBSCAN, graph based models such as the highly connected subgraphs, HCS, clustering algorithm, or neural models such as a neural network implementing a version of principal component analysis may be used. The method further comprises training 102 a hybrid autoencoder for each cluster of UEs. An autoencoder is an artificial neural network used to learn efficient codings for data compression of unlabeled data. Typically, autoencoders are used for data compression in the dimension domain. A trained autoencoder comprises two parametrized families of functions: a first family comprising functions which encode/compress data, collectively referred to as the encoder, and a second family of functions which decode/decompress data, collectively referred to as the decoder. For the purpose of the present disclosure, the encoder thus comprises functions which map CSI to compressed CSI, and the decoder comprises functions which map compressed CSI to reconstructed CSI. Training the autoencoder comprises the autoencoder learning suitable parameters for the encoder and the decoder. The person skilled in the art will note that the reconstructed CSI may not be identical to the original CSI. An autoencoder thus typically performs lossy data compression. Training the autoencoder comprises determining parameters of the encoder and of the decoder such that the data loss between the CSI and the reconstructed CSI is minimized. In applications, the person skilled in the art would strive for a compression rate sufficient to satisfy bandwidth requirements, while enabling as little data loss as possible. A hybrid autoencoder is a trained autoencoder comprising a family of functions adapted to be executed by quantum processing circuitry and a family of functions adapted to be executed by classical processing circuitry. For the purposes of the present disclosure, a hybrid autoencoder comprises a quantum neural network which, when trained, results in what may be referred to as a classical encoder and what may be referred to as a quantum decoder. In the description, the word encoder refers to the classical encoder and the word decoder refers to the quantum decoder. The word classical is used to indicate that a classical/conventional/von Neumann computing device is intended. The word “classical” may be interpreted to mean “non- quantum”. A classical computing device may, in embodiments, be a virtual computing device. A classical algorithm is an algorithm intended to be implemented by a classical computing device. The word quantum similarly indicates that a computing unit is comprised of some quantum processing circuitry acting on qubits or more generally qudits, and a quantum algorithm is an algorithm intended to be implemented on a quantum computing device. The person skilled in the art will note that the methods disclosed herein can be used for quantum computing devices for qudits, but qubits are used as an example throughout the detailed description for consistency and ease of notation in mathematical formulae. The quantum neural network is trained on training data comprising at least CSI data points corresponding to at least a selected UE of the cluster of UEs. The selected UE may be any UE in the cluster. In some embodiments, the selected UE may be the centroid of the cluster of UEs as determined using the measure of similarity. In other embodiments, the selected UE may be a centroid as determined using a different measure of similarity based on some property of the UEs of the cluster of UEs. In other embodiments, the selected UE may be selected at random from the cluster. In some embodiments, CSI from a subset of the UEs of the cluster, the subset including the selected UE, may be used as training data. The CSI corresponding to the selected UE may be processed before being used as training data. In some embodiments of the disclosure, the training data may be processed simply to remove outliers and to shape the data into an input format associated to the autoencoder. In other embodiments of the disclosure, the training data may be encoded 105 as quantum states using a state preparation circuit. The state preparation circuit may implement basis encoding. The state preparation circuit may be selected based on the characteristics of the quantum processing unit, such as a decoherence limit of the quantum processing unit, and a number of qubits (or more generally, qudits) of the quantum processing unit implementing the method. In some embodiments, part of the training data may be encoded as quantum states while other training data may remain in a classical form. Training the hybrid autoencoder may further comprise passing 106 the encoded training data and a reference state representing the target number of qubits to an encoder unitary. The encoder unitary is a quantum circuit comprising logical quantum gates which may be represented as unitary matrices. A gate which acts on ^ qubits is represented by a 2^{^} × 2^{^} unitary matrix. Training the encoder comprises determining parameters for the logical quantum gates. A single qubit |^^ may be represented as a vector

By way of example, a Pauli-X gate (the quantum equivalent to a NOT gate for classical computers) maps |0^ to |1^ and vice versa, corresponds to the unitary matrix 1

^ 0 and acts on a qubit represented as a vector by matrix multiplication. Letting a Pauli-X gate act on the qubit |^^ results in a qubit which may be represented as a vector as ^ ^ _^ ^_^ ^. Geometrically, the Pauli-X gate may be visualized as acting on the qubit by rotation by ^ radians around the x-axis of a Bloch sphere. The reference state represents the target number of qubits desired to compress the encoded data to. For example, the target number of qubits may be determined based on the dimension of the raw training data, the desired compressed dimension, and the encoding scheme used in the decoder. The desired compressed dimension may be determined by considering the available bandwidth for uplink CSI transmissions. Alternatively or in addition, the desired compressed dimension may be determined by weighing the available bandwidth against the desired accuracy of the recovered CSI at the network node. By way of example, if the compression ratio to be achieved is 0.75, that corresponds to the desired dimension of the compressed ^ data being ^ of the dimension of the CSI, and a binary encoding scheme where 4 qubits are used to encode each element of the CSI containing 4 elements, then the dimension of the reference state should be 4. Table 1 gives the expression to compute the size of the reference state for 4 common encoding schemes. In Table 1, the column dim(|Ψ_^^) contains the dimension of encoded training data point ^, dim(|Ψ^ ^ ^) contains the desired compression dimension, and dim (|^^) contains the corresponding dimension of the reference state |^^. The rows correspond to the encoding schemes binary encoding, angle encoding, dense angle encoding, and amplitude encoding. The parameter ^ denotes the number of qubits used to encode each element of the training data point, the parameter ^ denotes the number of elements in each training data point, and the parameter ^^ denotes the compression ratio to be achieved. Encoding Scheme dim⁽|^_^ ^{^)} dim⁽|Ψ_^ ^{^^)} _{= dim} ⁽ _|^ ^{^)} Binary ^ ∗ ^ ⌈^ ∗ ^ ∗ ^^⌉ ⌈^ ∗ ^ ∗ (1 − ^^)⌉ A^ngle _^ ^⌈ _{^ ∗ ^^} ^{⌉ ⌈} _{^ ∗ (1 − ^^)} ^⌉ Dense Angle ⌈^/2⌉ ⌈^/2 ∗ ^^⌉ ⌈^/2 ∗ (1 − ^^)⌉ Amplitude ⌈^^^_^^⌉ ⌈^^^_^^ ∗ ^^⌉ ⌈^^^_^^ ∗ (1 − ^^)⌉ Table 1 Training the hybrid autoencoder may further comprise using the training data to train 107 the encoder unitary to maximize the fidelity between trash states, wherein the trash states are the output states of the qubits traced out of the encoder outputs of the encoder unitary and the reference state. Training the hybrid autoencoder using the training data to train the encoder unitary may comprise using an update rule to update the parameters of the quantum logic gates of the encoder unitary. The update rule may be selected such that updating the parameters according to the update rule may, with high probability, lead to increased fidelity between the current measured trash state of the encoder unitary and the desired reference state for the encoder unitary. For the purposes of the present disclosure, the desired reference state comprises

where ^ represents the number of encoder qubits whose output state is forced to the |0^ state. An update rule to update the parameters of the quantum logic gates may be derived as follows. Let ^ = { ^_^^^ , ^_^^^} be the set of parameters for the autoencoder, with ^_^^^ denoting the set of parameters for the encoder unitary and ^_^^^ denoting the set of parameters for the decoder unitary. The input to the classical encoder is ℎ^{^} and the reconstructed CSI ℎ^{^^} at the quantum decoder is modelled by:

In Eq.1, the reconstructed CSI is obtained by applying a classical encoder function ^_^^^ dependent on the encoder parameters to the input data point at the UE. The encoder function is a classical encoder with a classical output. The classical output is received by the network node, where it is encoded into a quantum state by the state preparation function ^_^^^ , which hence takes a classical input and outputs a quantum encoding of the input. The decoder unitary ^_^^^ is then applied to the output of the state preparation function, which is dependent on the decoder parameters. Finally, the output of the decoder is measured by ^_^^^ to obtain the reconstructed CSI. The person skilled in the art will appreciate that this is a model of an ideal situation, not considering the effect of the channel on the received data at the network node. For the sake of clarity, the combination of the state preparation circuit, the decoder unitary, and the measurement function will be denoted ^_^^^ , which hence is a function representing the entire quantum decoder circuit, taking as input classical compressed CSI and providing as output a classical reconstruction of the CSI. A mean squared error over all the ^ training samples in the set of training samples may be used as a loss function. The loss function can thus be defined as:

where ^‖⋅^‖ _^ is the standard ^_^-norm. The loss may be minimized by gradient descent, updating each parameter of ^ at iteration ^ using the standard gradient descent update rule parametrized by a step size ^:

The partial derivative of the loss function may be rewritten as follows:

Using the chain rule, the partial derivatives of the encoder and decoder parameters can be expanded further:

This allows an expanded expression for the loss function as:

and hence an expanded update rule for the parameters is given by:

Eq.8 includes a derivative of the unitary encoder circuit. There are several methods for the person skilled in the art to evaluate a quantum gradient, one being a classical linear combination of the unitary matrices representing the quantum gates and another method of evaluation being a parameter shift method. The person skilled in the art may, alternatively, evaluate the gradient analytically rather than exactly by performing classical simulation. Once the gradient is calculated, the parameters may be updated and the loss function evaluated on the updated parameters to see if additional training is needed. The training may be considered complete when a stopping criterion is met. The stopping criterion may be, for example, that a rolling average of the output of the loss function is below a predetermined threshold. Alternatively or in addition, the stopping criterion may be that a predetermined number of iterations has been performed, or that an amount of computing resources has been used. A combination of stopping criteria may be used, where the training is complete when one stopping criterion is met. Alternatively, a loss function based on the average fidelity of the quantum decoder state and the encoded input state |Ψ_^^ may be used. Note that the encoded input state may be written as

A loss function can thus be defined as:

Using Eq.3 as a parameter update rule, we obtain a partial derivative of the loss function as: Substituting Eq.10 into the parameter update rule, we obtain an expanded parameter update rule based on the average fidelity of the quantum states as:

In Eq.11, |Ψ^^ is the equivalent quantum state of the compressed CSI data point ℎ^ ^_^ obtained from the classical encoder. The inner product of |Ψ^{^} _^^ and |Ψ_^^ is denoted

The method may further comprise obtaining 108 a decoder unitary by taking a complex conjugate of the encoder unitary. The encoder unitary can be represented as a series of unitary matrices acting on input vectors, or, equivalently, as a single unitary matrix. Taking a complex conjugate of the encoder unitary thus comprises determining a conjugate transpose, also known as a Hermitian transpose, of the matrix representing the encoder unitary. The resulting complex conjugate matrix can then be decomposed using, for example, unitary decomposition to obtain a set of quantum logic gates corresponding to a trained encoder circuit. The method may further comprise obtaining 109 the encoder by reducing the trained quantum encoder to a classical feedforward neural network. The quantum encoder may be reduced using tensor networks to obtain a neural network which may be executed by a classical processing unit at the UE (e.g. Rieser, H.-M., Köster, F., Raulf, A. P. “Tensor networks for quantum machine learning”, ArXiV 2302.11735v1). The reduced trained encoder is light-weight and can be used by a UE without quantum computing abilities. The method further comprises transmitting 103 each encoder of the trained hybrid autoencoder to each UE of the associated cluster of UEs. That is, each trained autoencoder corresponds to a cluster of UEs. The encoder part of the autoencoder is transmitted to each UE in the cluster of UEs. Obtaining the encoder part of the trained hybrid autoencoder enables the receiving UEs to compress the CSI before being transmitted to the network node. The compressed CSI can be decoded at the network node using the quantum decoder of the hybrid autoencoder. The encoder may be transmitted using any communication channel between the network node and the UE. An encoder, or representation of an encoder, may be transmitted using for example existing model formats such as Open Neural Network Exchange (ONNX), or formats used in commonly used toolboxes such as Keras or Pytorch. In general, a machine learning model can be transmitted using a high-level model description, potentially together with more detailed information. High-level model description (model parameter vector) may for example comprise parameters defining the structure and characteristics of the model, such as for example number of layers, number of nodes per layer, activation function of respective layer, nature of connections between nodes of respective layer, just to mention a few examples of hyperparameters. Alternatively, a model may be transmitted using container-based signalling, for example using a containerized image. In some embodiments of the disclosure, the method of the invention is performed in a network node other than the network node in direct communication with the UE, and thus transmitting the encoder to the UE may comprise transmitting it to a second network node such as a radio base station or a Wi-Fi access point and the second network node transmitting the encoder to the UE. In embodiments where the method is performed by a node other than an access node, the decoding may be performed either by the node which trains the autoencoder, or the decoder may be transmitted to the access node for the access node to use to decode the CSI. The method may further comprise finetuning 104 each autoencoder at predetermined events on training data corresponding to a subset of UEs in the corresponding cluster of UEs. The predetermined events may be, for example, predetermined points in time, times when the transmission quality drops below a predetermined threshold, or times when a behavior of the UEs, such as moving speed relative to the network node, changes in a substantial way. In general, the predetermined events may be events that indicate that the performance of the beamforming module in the network node has degraded or is expected to degrade below some threshold. The finetuning is performed at the network node. After finetuning the autoencoder, the updated encoder is transmitted to the UEs. Fine-tuning each autoencoder may comprise finetuning the parameters of each hybrid autoencoder using training data obtained by mixing training data obtained from each of the encoders of the UEs in the cluster except for training data from the selected UE from that cluster. That is, the finetuning process comprises obtaining additional training data from the UEs in the cluster of UEs and using the additional data as training data for further training of the autoencoders. Since the UEs have access to a trained encoder, the training data may be compressed by the UE using the encoder and transmitted to the network node performing the finetuning. Training data from the selected UE, which was used to train the encoders, may in some embodiments be excluded from the training set used for finetuning. The training data from the remaining UEs may be mixed, so the finetuning process may encounter data from multiple UEs without having to train on all available data. In other embodiments, training data from the selected UE is included in the mixed data used for finetuning. In some embodiments, only data from UEs on which the hybrid autoencoder performs below a predetermined threshold is used for finetuning. In these embodiments, the purpose of the finetuning may be to adapt the trained autoencoder to a subset of the UEs in the cluster on which the autoencoder performs particularly poorly. Therefore, training data only from these UEs is used to finetune the autoencoder. Fig.2 depicts a handshake diagram of exemplary communications between a network node 200a and a UE 200b for CSI reporting. The process for CSI reporting comprises the access node transmitting 201 a reference signal to the UE. The reference signal may be defined according to a suitable standard. The UE then computes 202 CSI from the reference signal. Computing the CSI may further comprise preparing the CSI for transmission. In some embodiments, this may comprise using the trained encoder to compress the calculated CSI for transmission. In some embodiments of the disclosure, the CSI comprised in the training data is obtained by the network node receiving CSI compressed at the UE via random projections. The person skilled in the art may note that, for the finetuning process, the additional training data may be transmitted using the trained autoencoder. However, for the original training, this is not an option. Moreover, for the finetuning process, this may be undesirable since the finetuning may not be triggered until after the performance of the autoencoder has degraded, rendering training data obtained by means of the autoencoder unreliable. In embodiments, this problem may be resolved by the UEs compressing potential training data by means of random projections. Compression by means of random projections may comprise obtaining a CSI matrix ^ of some dimension ^ × ^. Compressing the CSI matrix then comprises reshaping the CSI matrix into an ^^ × 1 dimensional vector ℎ. The UE may have access to a matrix Φ of dimension ^ × ^^ where ^ is the desired size of the compressed CSI and the elements of Φ are randomly generated offline based on Gaussian distributions. Both the base station and the UE have access to the same matrix Φ. The matrix Φ may thus be generated offline and provided to the UE and base station. Using the matrix Φ to compress the CSI comprises calculating a matrix product ^ = where ^ is an ^ × 1 dimensional vector which may be transmitted to the base station. Compression by means of random projections may take advantage of the fact that CSI data displays some predictability in the form of sparsity in the angular-delay domain as the received power at the base station is concentrated in a limited number of angles, and the time delay between multipath component arrivals lies within a limited period. Therefore, the UE may start to compress the CSI by using a two dimensional discrete Fourier transform to truncate the dimension of the CSI matrix to the most relevant parts of the frequency and time spectrum. After truncating the matrix, the UE may perform compression using random projections. The random projection compression thus comprises mapping a CSI parameter to a linear combination of orthogonal functions with complex weights. The UE then transmits 203 the compressed CSI as a CSI report. In some embodiments, this may comprise transmitting the compressed CSI obtained from the trained encoder. In other embodiments, the compressed CSI comprises the compressed complex coefficients comprised in ^ as part of the CSI report. The CSI report may, for example, be transmitted in an uplink control channel or a shared channel to the base station. At the base station, the received CSI must be reconstructed 204 to either be used for determining downlink parameter, or to be used for training or finetuning an autoencoder. The reconstruction may comprise using the trained decoder to decompress the received data. Alternatively, to reconstruct 111 the uncompressed, but possibly truncated, ℎ at the base station, the decoder at the base station solves the following L1-norm problem using, for example, a Quantum Alternating Operator Ansatz (QAOA) scheme on a quantum processing unit, or by using a quantum annealer: min^‖ℎ^‖ _^ such that Φℎ = ^

To transform the L1-norm problem to a linear program, define variables ^^{^}, ^^{^}, … , ^^{^^} where ^^{^} is the upper bound of

Then the problem can be reformulated as: ^^ min ^ ^^{^} such that Φℎ = ^, ^ℎ^{^}^ ≤ ^^{^} ∀^ = 1,2, … , ^^ ⁽^^.13⁾. ^^^ The constraints are equivalent to 2^^ linear constraints without absolute value signs. The linear problem may then be converted to a quadratic unconstrained binary optimization, QUBO, problem. The inequality constraints are converted into equality constraints by adding slack variables ^^{^} , ^^{^} to obtain constraint equations: ℎ^{^} + ^^{^} = ^^{^} , −ℎ^{^} + ^^{^} = ^^{^} ∀^ = 1, 2, … , ^^. The variables ℎ^{^}, ^^{^} , ^^{^} and ^^{^} can be expressed as binary variables using

Replacing the variables with the binary expansion expression results in constraint equations written on the form:

The equality constraint Φℎ = ^ is expanded as follows:

or, more compactly:

Replacing ℎ^{^} with the equivalent binary expansion form results in the expanded form of Eq.15:

Replacing ^^{^} in the objective function results in:

The equality constraints are added as a quadratic penalty term (^^ − ^)^{^} where ^ is a matrix comprising the coefficients of the binary variables in column vector ^ and ^ is a column vector comprising the constants in the system of linear equations ^^ = ^ obtained from the constraint equations above. An energy function to be minimized is thus: ^ ≈ ^^{^[}^ + ^^^{^}^ − ^^^^⁽2^^^{^}^^)]^ ⁽Eq.16⁾ where ^ is a penalty term such that the penalty for violating a constraint is larger than the value of the objective function introduced to ensure no constraint will be violated by the probabilistic nature of the quantum processing circuitry used to recover the CSI. The binary variables in the problem represent the logical qubits in the problem graph which may be mapped to a Chimera graph of the quantum processing unit through the embedding process. The mapping of the binary variables to logical qubits is given by:

modulo operation. Each equality constraint corresponds to a row in the ^ matrix. The row elements corresponding to the ^^{^^} constraint:

are given by: ^_^,^ = 2^{^} ∀ ^ ∈ ^(^ − 1) ∗ ^_^1, .. , ^ ∗ ^_^^, ∀ ^ ∈ ^0,1, .. , ^_^ − 1^

∈ ^0,1, .. , ℎ_^ − 1^ ^_^,^ = −2^{^} ∀ ^

∀ ^ ∈ ^0,1, .. , ^_^ − 1^ (^^.18). Similarly, the row elements corresponding to the ^^{^^} following constraints:

are given by: ^_^,^ = 2^{^} ∀ ^ ∈ ^{^(}^ − 1⁾ ∗ ^_^1, .. , ^ ∗ ^_^ ^{^}, ∀ ^ ∈ ^{^}0,1, .. , ^_^ − 1^{^}

∈ ^{^}0,1, .. , ℎ_^ − 1^{^} ^_^,^ = −2^{^} ∀ ^

+ ^^ ∗ ^_^ + ^ ∗ ^_^^ (^^.19). Finally, the row elements corresponding to the ^^{^^} following constraint:

are given by:

The QUBO formulation obtained above can be embedded into the quantum processing unit or quantum annealer at the base station using, for example, minor graph embedding and the output will be a reconstruction of the original CSI which may be used for training the autoencoder. Alternatively or in addition, the reconstructed CSI may be used to finetune the autoencoder. In some embodiments, the network node comprises one set of quantum processing circuitry which may be used both to train the autoencoder, reconstruct data using the trained decoder, and reconstruct training data using e.g. a QAOA scheme. Optionally, the network node may additionally comprise a quantum annealer for reconstructing training data using a quantum annealing process. The quantum annealer comprises quantum circuitry suitable for performing a quantum annealing optimization process to solve the QUBO problem. Finally, the base station may use the reconstructed CSI to either train or finetune an autoencoder, or to determine parameters for downlink transmission and transmitting 205 data to the UE. Fig.3 depicts an exemplary flow of communication in a network according to an embodiment of the disclosure. The flow of communication is depicted between a network node 300a and a UE 300b and relates to the distribution of the trained encoder and updating of the trained encoder. The flow of communication is initiated by the network node training 301 an autoencoder according to an embodiment of the present disclosure, and obtaining from the autoencoder a classical encoder. After obtaining the encoder, the network node may transmit 302 the encoder to the UE. Transmitting the encoder to the UE may be performed using any available channel of the established communication channel between the network node and the UE. In embodiments, the encoder may be transmitted by the network node using radio resource control, RRC, signaling. Alternatively, in some embodiments, the encoder may be transmitted as part of CSI- RS signaling in the downlink. After the encoder has been communicated to the UE, the UE can compress and transmit CSI, using the encoder, to the network node, which the network node may decode with the decoder and use for beamforming for future communication with the UE, from control channel signaling to data traffic transmissions. After some time, the network node may detect 303 an event indicating that the performance of the autoencoder has degraded below a certain threshold. Detecting such an event may be considered a trigger for the finetuning process to begin. For the finetuning process, the network node requires additional training data. The additional training data may be training data which was not used for the initial training process, i.e. training data from UEs in the cluster of UEs which was not used in the initial training process. Alternatively, the network node may request 304 training data, in the form of CSI, from some or all of the UEs in the cluster of UEs. Requesting training data may comprise transmitting a reference signal to each UE in the cluster of UEs. In some embodiments, the network node may specify how the UE is to respond, i.e. whether the UE should prepare 305 a CSI report using the existing encoder, or using random projections, or using some other method. The CSI report can then be transmitted 306 by the UE to the network node to be used as training data for the finetuning process. Finetuning 307 the autoencoder comprises using the training data, or some subset of the training data, to continue training the parameters of the autoencoder. Once some stopping criterion has been met by the finetuning process, the network node may transmit 308 the finetuned parameters comprising the updated encoder to the UE. In embodiments where the training of the autoencoder happens in a network node other than the access node corresponding to the set of UEs, the network node may further transmit the decoder to the access node. Fig.4 depicts a network node 400 which may perform a method according to an embodiment of the disclosure. The network node may in embodiments be a node in a telecommunication network. In some embodiments, the network node may be a radio access node, such as an eNodeB, a gNodeB, or a radio access node according to any future standard. The network node may comprise a memory 404, and processing circuitry 405. The processing circuitry may comprise radio frequency transceiver circuitry 412 and baseband circuitry 414. The processing circuitry may further comprise quantum processing circuitry 424 on which an autoencoder according to embodiments of the disclosure may be trained and/or on which a decoder according to embodiments of the disclosure may be implemented. The processing circuitry may further comprise quantum processing circuitry 426 on which a decoder according to embodiments of the disclosure may be implemented. The processing circuitry may further comprise quantum processing circuitry 428 on which a quantum annealer according to embodiments of the disclosure may be executed. The network node may further comprise a power source 408. The network node may further comprise a communication interface 406. The communication interface may further comprise radio front-end circuitry 418, the radio front-end circuitry further comprising a filter 420 and an amplifier 422. The communication interface may further comprise a port or a terminal 416 and an antenna 410. In embodiments of the disclosure, the antenna may be a MIMO arrangement of antenna elements. Fig.5 depicts a logic representation of quantum processing circuitry corresponding to an autoencoder according to an embodiment of the disclosure. The autoencoder operates on an input set of ensemble states that lie in a composite Hilbert space ℋ_^^ ⊗ ℋ_^ ^{^} where ^|Ψ_^ ^{^} ∈ ℋ_^^ and ^|^^{^} ∈ ℋ_^ ^{^}. The autoencoder circuit is initialized with the input state |^^^ =

⊗ |^^_^ ^{^}. The initial density matrix ^_^^ is defined as

The circuit first acts only on the subspace ℋ with t ^ ^_^ he unitaries ^_^^ whose parameters can be optimized and then the circuit measures the trash state on the ℋ_^ subspace. Next, the circuit acts with unitaries ^ ^{^ ^} ^^^{^} only on the subspace ℋ_^^ ^{^}, using the ^ reference qubits of |^^{^} in ℋ_^ ^{^} to reconstruct the initial state. The output density matrix ^_^^^ is given

partial trace function on the ℋ_^ subspace. The unitaries in Fig.5 can be expressed as ^ = ^^{^^}, for some Hermitian matrix ^. Such matrices may be written as linear combinations of tensor products of Pauli matrices and the identity matrix. A general ^-qubit unitary may be expressed as ^_^ =

⊗ … ,⊗ where ^_^are the Pauli matrices for ^ ∈ {1,2,3} and ^_^ is the 2 × 2 identity matrix. The parameters ^_{^^,…,^^} are trained by a learning rule according to an embodiment of a method of the disclosure. Fig.6 depicts a UE 600 according to an embodiment of the disclosure. The UE may comprise processing circuitry 602, a bus 604, and input/output interface 606, and a power source 608. Th UE may further comprise a memory 610 and a communication interface 612. The memory may further comprise application programs 614 and data 616. The communication interface may further comprise a transmitter 618 and a receiver 620. The UE may further comprise an antenna 622. The processing circuitry may comprise processing circuitry suitable for performing methods according to the disclosure. The memory may comprise machine-readable instructions 430, which, when executed by the processor, cause the UE to perform methods according to the disclosure. The memory may comprise a computer program product 428 comprising computer readable instructions 430. Fig.7 depicts a communication system 700 in which methods according to embodiments of the disclosure may be performed. The communication system may, for example, be a telecommunication system or a communication system incorporating Wi-Fi communication. In embodiments, the communication system may be a hybrid system incorporating multiple wireless communication standards. The communication system may comprise a host 701 and a telecommunication network 702. The telecommunication network may comprise an access network 704 and a core network 706. The core network may comprise one or more core network nodes 708. The access network may comprise one or more network nodes 710A, 710B. The network nodes may be in communication with UEs 712A, 712B. Alternatively or in addition, the network nodes may be in communication with a hub 714, where the hub communicates with UEs 712C, 712D.

Claims

CLAIMS 1. A method (100) for enabling compression of channel state information, CSI, the method performed by a network node (400) of a communication system (700), comprising: clustering (101) user equipment, UEs, (600) associated to the network node (700) according to a measure of similarity; training (102) a hybrid autoencoder for each cluster of UEs, the hybrid autoencoder comprising a classical encoder and a quantum decoder, on training data, the training data at least comprising CSI data points corresponding to a selected UE in the cluster of UEs; transmitting (103) each encoder of the trained hybrid autoencoder to each UE of the associated cluster of UEs.

2. The method (100) according to claim 1, wherein the measure of similarity is based on one or more of: deterministic multipath components of a signal between each UE and the network node; beam indices of a codebook of the network node, wherein the beam indices correspond to beams in a pre-defined beam grid associated to an established channel between each UE and the network node; and a channel covariance matrix of an established channel between each UE and the network node.

3. The method (100) according to claim 1 or 2, wherein training (102) the hybrid autoencoder comprises: encoding (105) each training data point in the training data set as a quantum state using a state preparation circuit implementing basis encoding; passing (106) the encoded data point and a reference state representing the target number of qubits desired to compress the encoded data point to an encoder unitary; and training (107) the encoder unitary to maximize the fidelity between trash states of the encoder unitary and the reference state; obtaining (108) a decoder unitary by taking a complex conjugate of the encoder unitary; and obtaining the encoder (109) by reducing the trained quantum encoder to a classical feedforward neural network.

4. The method (100) according to claim 3, wherein the size of the reference state is computed from the dimension of the training data points, a desired compression ratio, and an encoding scheme used in the decoder unitary.

5. The method (100) according to any of claims 3-4, wherein a loss function used to train both the classical encoder and the quantum decoder is a function of the data encoded in quantum states.

6. The method (100) according to any of claims 3-5, wherein a loss function used to train the classical encoder is a function of the training data before encoding the training data into quantum states and a loss function used to train the quantum decoder is a function of the training data encoded in quantum states.

7. The method (100) according to any of claims 1-6, wherein the selected UE for each cluster of UEs is the centroid of the cluster of UEs.

8. The method (100) according to any of claims 1-7, further comprising at predetermined events, finetuning (104) each trained hybrid autoencoder on training data corresponding to a subset of UEs in the corresponding cluster of UEs.

9. The method (100) according to claim 8, wherein fine-tuning (104) the hybrid autoencoder comprises finetuning the parameters of the hybrid autoencoder using training data obtained by mixing training data from each of the encoders of the UEs in the cluster except for training data of the selected UE.

10. The method (100) according to claim 9, wherein the training data further comprises test batches of data from the encoder of the selected UE.

11. The method (100) according to any of claims 8-10, wherein only data from UEs on which the hybrid autoencoder performs below a predetermined threshold is used for finetuning.

12. The method (100) according to any of claims 8-11, wherein finetuning is triggered by a change in the moving speed of a subset of the UEs.

13. The method (100) according to any of claims 1-12, further comprising obtaining the CSI comprised in the training data by receiving (110) CSI compressed at the UE via random projections and reconstructing (111) the CSI at the network node by solving an L1-norm optimization.

14. A network node (400) of a communication system (700) for enabling compression of channel state information, CSI, the network node comprising a memory and processing circuitry, the processing circuitry comprising a quantum processing unit, the network node configured to: cluster user equipments, UEs, associated to the network node according to a measure of similarity; train a hybrid autoencoder for each cluster of UEs, the hybrid autoencoder comprising a classical encoder and a quantum decoder, on training data, the training data at least comprising CSI data points corresponding to a selected UE in the cluster of UEs; transmit each encoder of the trained hybrid autoencoder to each UE of the associated cluster of UEs.

15. The network node (400) according to claim 14, further configured to perform a method according to any of claims 2-13.

16. The network node (400) according to claims 14 or 15, where the network node is a radio access node.

17. A computer program (430) comprising machine-readable instructions which, when executed on the processor of a network node, cause the network node to perform a method according to any of claims 1-13.

18. A computer program product (432) comprising a computer program according to claim 17.