A quantum convolutional autoencoder

Abstract

A quantum computing device (200) for use as a convolutional filter acting on a zeroth, first and second qubit (qo, qi, q2), comprising: a first rotation block (210) configured to perform one or more phase rotations on the zeroth qubit conditional on the first qubit; an entangling block (220) arranged immediately downstream of the first rotation block and configured to provide entanglement between the first and second qubits; and a second rotation block (230) arranged immediately downstream of the entangling block and configured to perform one or more RY rotations on the zeroth qubit conditional on the first qubit. A quantum convolutional autoencoder, QCAE, acting on n input qubits and having a compression rate of Rc, comprising a sequence of NL layers of cascaded instances of the quantum computing device (200) and NL layers of cascaded instances of an inverse of said quantum computing device

Description

A QUANTUM CONVOLUTIONAL AUTOENCODER TECHNICAL FIELD

[0001] The present disclosure relates to the field of data encoding, notably in data encoding for digital communication networks. It proposes quantum computing devices for use as convolutional filters and autoencoders, as well as techniques for training such quantum computing devices. BACKGROUND

[0002] An autoencoder, in the conventional sense, is an unsupervised neural network that learns how to efficiently compress data, a process also called encoding. The autoencoder also learns how to reconstruct the data from the compressed representation in such manner that the difference between the original data and the reconstructed data is minimal. Applications of Auto-Encoder in Wireless Communication Systems

[0003] Autoencoders are increasingly finding their applications in wireless communication systems. For example, instead of optimizing each component (e.g., channel coding, modulation, equalization, synchronization etc.) independently, an autoencoder can provide better performance by jointly optimizing the transmitter and the receiver [1,2]. Similarly, in frequency division duplex (FDD) mode of massive multiple-input multiple-output (MIMO) systems, the channel state information is efficiently exchanged between the user equipment and base station by combining the learning of a sparse feedback format with the task of performing channel reconstruction at the base station using an autoencoder neural network [3]. They are also used in spectrum sensing, where they learn features from the RSSI (received signal strength indicator) measurements, and the subsequent classifier determines whether the spectrum of input samples is occupied by a primary user or a secondary user [4].

[0004] In the network, they are used for dimensionality reduction and the reconstruction error is utilized to classify anomalies in network intrusion [6]. The features selected through auto-encoders are also used detect anomalies on network performance [7] datasets. They are used to model uncertainty in network flows by capturing prior distribution [9].

[0005] Auto-encoders are employed on as nonlinear feature extraction scheme on customers data collected from telecom operators to enhance the performance of models predicting customer churn [5]. They are also used to extract latent representations that can improve the detection of telecom fraud [8]. Quantum Autoencoder

[0006] A quantum autoencoder (QAE) may consist of a pair of parametrized quantum circuits, one that compresses the information in ^ input qubits onto ^ latent space qubits (encoder circuit) and one that decompresses it back into ^ qubits (decoder circuit). The circuits are trained to encode and decode a specified set of training input data using machine learning methods.

[0007] The circuit 400 shown in figure 4 illustrates a QAE originally disclosed in

[0011] . It operates on an input set of (ensemble) states that lie in a composite Hilbert space ℋ^^⊗ ℋ^^, where|^^^∈ ℋ^^and|^^∈ ℋ^^. It is initialized with the input state |^^^= |^^^^^⊗ |^^^^. The initial density matrix ^^^is equal toThe circuit first acts only on the ℋ ^ ^^subspace with the unitaries ^^^, whose parameters are to be optimized, and then measures on the ℋ^subsystem. Next, it acts with the unitaries ^ ^^ ^^^(adjoint of ^ ^ ^^ ) only on ℋ^^^requiring the ^ reference qubits |^^in ℋ^^to reconstruct the initial state. The output density matrix ^^^^is equal towhere Tr^is a partial trace function on the ℋ^subsystem.

[0008] The unitaries in figure 4 can be expressed as ^ = ^^^, where ^ is a Hermitian matrix. Such matrices can be written as linear combinations of tensor products of the Pauli matrices and the identity. A general N-qubit unitary can be expressed aswhere ^^are the Pauli matrices for ^ ∈{1,2,3}and ^^is 2x2 identity matrix. The parameters ^^^,…,^^are trained by the learning rule and could potentially be way less than the weights in classical autoencoders and thus the training load could be smaller for the quantum autoencoders.

[0009] There exist approaches describing varying types of quantum autoencoders, using varying architectures and investigating theoretical limits of such models. One architecture contains multi-qubit gates acting on all possible pairs of the ^ qubits, whereas the second one contains all possible controlled general single qubit gates on the n qubit register, plus single qubit rotation gates in the beginning and end.

[0011] . The two proposed architectures are parameterized correspondingly by 15^(^ − 1) / 2 and 3^(^ − 1) + 6^ parameters.

[0010] There exists a theoretical proof

[0019] that QAE can be used to efficiently calculate eigenvalues and prepare corresponding eigenvectors of a high-dimensional quantum state with the low-rank property. Another study

[0012] suggests that, for perfect quantum autoencoders, first the trash state must be completely disentangled from the lower dimensional representation of the input state (in the latent space) and always taking a fixed reference state value. Second, the maximum number of linearly independent vectors in the input states should not be more than (or is equal to) the dimension of the target latent space. Given these conditions, an upper bound of the compression rate for quantum autoencoders is proven to be the maximum sum of partial eigenvalues of the density operator representation of the input states. Moreover, if a quantum autoencoder can compress a specific set of input states with perfect fidelity (between the trash state and reference state) using the same unitary U, it can compress any linear combination of those states with the same level of fidelity.

[0011] For training, QAEs propose to swap the trash states with a reference state, then measure the inner product of the trash states with the reference states as the cost function and apply a gradient-descent solver

[0011] . For cases where it does not have perfect fidelity with the reference state, the spectra of the trash state density matrix are extracted using variational quantum diagonalization and incorporated into the loss function

[0014] . One can also include a feature vector into the variational quantum circuit a that characterizes the model to be compressed. The encoder unitary then has additional dependence on this feature vector, which tends to increase the fidelity once trained

[0016] .

[0012] However, one study

[0013] shows that for a broad class of variational quantum circuits, the range of variation of the cost function, when this variation is provided via adjusting any local quantum gate within the circuit, vanishes exponentially with the number of qubits. The observation in

[0013] may rule out training strategies for VQAs that i) suffer from vanishing gradients, ii) optimize local unitaries independently, even based on gradient-free methods, and iii) use reparameterization within local unitaries. There also exists a quantum annealing based approach

[0018] , which attempts to find a set of basis vectors and their binary coefficients that precisely reconstruct each input vector in the original data. The optimization for the basis vectors is performed classically, while binary coefficients are retrieved through quantum annealing. In other words, it compresses classical input data onto a classical latent space but uses quantum annealing for the minimization of the compression loss function. It also suggests a bias correction procedure to estimate and eliminate the error and bias introduced from the inexact reconstruction of the lossy compression for statistical data analysis. Quantum Convolution Auto-Encoders

[0013] A potentially promising and not yet well-explored category of quantum encoders is quantum convolutional autoencoders (QCAE). An encoder in this category contains convolutional layers made up of convolution filters (i.e., circuit blocks that act on a subset of the input qubits at a time) which are intended to compress the input data of the subsets of qubits the filters act on. It also contains pooling layers, where some of the input qubits (the ones whose information has been transferred to other qubits and which have ideally been put in a predetermined state such as the |0> state by the circuit) are measured and / or reset. The encoder typically consists of a sequence of several convolutional layers and at least one pooling layer, decreasing the latent space size step by step.

[0014] Prior art shows that the additional of convolution or “quanvolutional” layers make quantum convolutional neural networks (QCNN) resistant to exhibit barren plateaus as the gradient vanishes only polynomially with the number of qubits

[0021] and the QCNN are trainable under random initialization. Here the QCNN consists of a “quanvolutional” layer with 2-qubit gates between each pair of adjacent qubits, a pooling layer implemented by having pairs of qubits being mapped onto single qubits, namely by measuring the first qubit and performing a gate on the second qubit controlled by the classical information of the measurement. Note that this architecture is different from the quantum augmented convolutional network

[0022] , where the “quanvolutional” layer consisting of state encoding, parameterized random quantum circuit and measurement operation replaces the classical convolution layers and show that proposed.

[0015] To reap the benefits of “quanvolutional” layers in autoencoders, a study

[0017] investigates parametrized randomized quantum circuits as quantum convolutional autoencoders, analogous to classical convolutional autoencoders. Here classical data is encoded into qubits using amplitude encoding, followed then by the unitary evolution, under the “non-trainable” quantum filtering layer and then measuring all the qubits and saving the data in a classical latent space. Here this layer is pre-computed prior to the training of the network.

[0016] The existing research in this technical field has not covered quantum autoencoders for compressing information encoded using so-called Dense Angle Encoding (DAE), where each qubit stores two classical continuous variables. For a feature vector ^ =[^^, … ^^]^∈ ℝ^, a dense angle map ^ ↦ ^(^) can be defined asThe first layer in quantum convolutional autoencoders can process the information encoded using angle encoding.

[0017] The prior art does not cover quantum convolutional autoencoders with optimization of the parameters of convolutional filters during training phase. Prior art also does not cover modular (building-block) architectures for quantum convolutional autoencoders.

[0018] The study

[0011] makes the most general possible ansatzes for the circuits, which leads to the largest possible number of generalization parameters and large circuits. In use cases where it is desired to train an autoencoder for a more specific task, however, it may be hypothesized that choosing a more specific ansatz, such as an ansatz guided by physical understanding or characteristics of the task, may allowfor a more efficient quantum encoder. This way, efficiency may be gauged in terms of network size / depth and number of training parameters needed for a particular problem and desired error-free compression rate. SUMMARY

[0019] An objective of the present disclosure is to provide a convolutional quantum autoencoder specifically adapted to a telecommunications network.

[0020] In a first aspect of the present disclosure, there is provided a quantum computing device for use as a convolutional filter acting on a zeroth, first and second qubit q0, q1, q2, the quantum computing device comprising: a first rotation block, which is configured to perform one or more phase rotations on the zeroth qubit conditional on the first qubit; an entangling block, which is arranged immediately downstream of the first rotation block and configured to provide entanglement between the first and second qubits; and a second rotation block, which is arranged immediately downstream of the entangling block and configured to perform one or more ^^rotations on the zeroth qubit conditional on the first qubit.

[0021] The quantum computing device corresponds to an ansatz for a quantum convolutional filters which is adapted for dense-angle-encoded input. These ansatz can also compress dense-angle-encoded intermediate data between convolutional layers.

[0022] In a second aspect, there is provided a quantum convolutional autoencoder (QCAE) acting on ^ input qubits and having a compression ratio of ^^, the QCAE comprising a sequence of: ^^layers of cascaded instances of the quantum computing device with the features of the first aspect; and ^^layers of cascaded instances of an inverse device of said quantum computing device. In the QCAE, the number of layers is related to the compression ratio by ^^= log^ ^^; ^ ^ the first layer of the QCAE has ^ instances of said quantum computing device, where ^ is a multiple of 3; for each next layer of the QCAE, the number of instances of said quantum computing device is decreased by a factor 2 / 3 by resetting the zeroth outputqubit of each quantum computing device in the previous layer; and the ^^’th layer of the QCAE has at mostinstances of said quantum computing device.

[0023] In other words, the disclosure proposes a modular architecture, where the convolutional filters as combined as building blocks into a larger quantum con- volutional autoencoder with several layers, to achieve a desired compression ratio. It is notable that the QCAE can be implemented such that each convolutional filter contains a mere 23 gates and has a depth of 18 gates while compressing in a 3-to-2 ratio. This is significantly shorter than using a 3-to-2 filter made from generic two- qubit unitary circuits (each containing 18 gates and being 11 gates deep)

[0023] plus entangling block (totaling 40 gates, 26 gates deep), or using one general three-qubit unitary circuit (contains 138 gates)

[0024] . The reduced depth allows this QCAE to have relatively higher accuracy for noisy intermediate-scale quantum computers where depth affects the coherence directly.

[0024] In a third aspect of the present disclosure, there is provided a method of training a QCAE. The comprises: providing a QCAE with the features of the second aspect outlined above; defining a cost function ^^on the basis of a difference between, on the one hand, a z-spin measurement on all zeroth output qubits that are being reset and, on the other hand, a label corresponding to an outcome where all z-spin measurements are applied to a zero state; providing a set of dense-angle-encoding training vectors; and optimizing parameter values in the QCAE over the set of training vectors and with respect to the cost function, wherein each training vector represents a quantum state.

[0025] The proposed method allows more efficient training of the parametrized quantum computing device, which need not rely on the complete available training data set. Instead, the disclosure demonstrates that a subset of training data such that the corresponding quantum states are orthogonal to each other is sufficient. The minimum number of elements by which the Hilbert space under consideration can be spanned is equal to the dimension of that space.

[0026] In broad terms, the disclosure proposes an ansatz for a quantum convolutional filter that can be trained to achieve compression of dense-angle- encoded input data (and decompression using the inverse of the filter) with a high fidelity. The ansatz is designed to allow for connecting several such filters in parallel to act on a larger register of qubits and make up a convolutional layer. Several such layers can be connected in series to achieve a higher compression ratio equal to the number of layers to a power of 2 / 3. The decoder corresponding to this convolutional encoder consists of the exact inverse of it, which is possible since the encoder is a unitary quantum circuit. To reduce the training overhead, the disclosure generates a subset of training data such that the corresponding quantum states are orthogonal to each other and show that reconstruction error remains within the acceptable bound

[0027] The present disclosure further relates to a computer program containing instructions for causing one or more processors to carry out the above training method. The computer program may be stored or distributed on a data carrier. As used herein, a “data carrier” may be a transitory data carrier, such as modulated electromagnetic or optical waves, or a non-transitory data carrier. Non-transitory data carriers include volatile and non-volatile memories, such as permanent and non- permanent storage media of magnetic, optical or solid-state type. Still within the scope of “data carrier”, such memories may be fixedly mounted or portable.

[0028] Generally, all terms used in the claims are to be interpreted according to their ordinary meaning in the technical field, unless explicitly defined otherwise herein. All references to "a / an / the element, apparatus, component, means, step, etc." are to be interpreted openly as referring to at least one instance of the element, apparatus, component, means, step, etc., unless explicitly stated otherwise. The steps of any method disclosed herein do not have to be performed in the exact order disclosed, unless explicitly stated. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Aspects and embodiments are now described, by way of example, with reference to the accompanying drawings, on which: figure 1 shows, as an illustration of dense angle encoding, an arrangement of quantum logic gates acting on three input qubits;figure 2 shows a block structure of a quantum computing device for use as a convolutional filter; figure 3A shows an example implementation of the quantum computing device of figure 2; figure 3B shows an example implementation of an inverse device of the quantum computing device of figure 2; figure 4 is a block diagram of a generic quantum autoencoder; figure 5A is an example implementation of a quantum convolutional autoencoder, which includes a sequence of layers of cascaded instances of the quantum computing device of figures 3A and 3B; figure 5B is an example implementation of the encoder part of a further quantum convolutional autoencoder; figure 6 is a flowchart of a method of training a quantum convolutional autoencoder; figure 7 is a plot of results obtained in Trial 1, showing cost function as a function of training step for learning rate = 0.01 and beta = 0.95; figure 8 is a plot of results obtained in Trial 1, showing cost function as a function of training step for learning rate = 0.001 and beta = 0.5; figure 9 is a plot of results obtained in Trial 2, showing cost function as a function of training step for learning rate = 0.01 and beta = 0.95; figure 10 is a plot of results obtained in Trial 1, showing cost function as a function of training step for learning rate = 0.001 and beta = 0.1; figure 11 is a plot of results obtained in Trial 3, showing cost function as a function of training step for learning rate = 0.01 and beta = 0.95; figure 12 is a plot of results obtained in Trial 3, showing cost function as a function of training step for learning rate = 0.001 and beta = 0.5; and figure 13 is a block diagram of a generic computing system for executing an embodiment of figure 6.DETAILED DESCRIPTION

[0030] The aspects of the present disclosure will now be described more fully hereinafter with reference to the accompanying drawings, on which certain embodiments are shown. These aspects may, however, be embodied in many different forms and should not be construed as limiting; rather, these embodiments are provided by way of example so that this disclosure will be thorough and complete, and to fully convey the scope of all aspects of the disclosure to those skilled in the art. Like numbers refer to like elements throughout the description. Encoding classical data using Dense Angle Encoding

[0031] Before describing the disclosed ansatz for a convolutional filter and how it can be connected in a modular form to quantum convolutional autoencoder, Dense Angle Encoding will be briefly introduced in addition to the already introduced mapping (1).

[0032] The classical input data into the convolutional filter is encoded onto the qubits using Dense Angle Encoding, where each qubit stores two classical continuous variables, one representing its amplitude and one in its phase). A quantum circuit with an arrangement of quantum logic gates that implements the Dense Angle Encoding is shown in Figure 1, which is configured for three input qubits (^0, ^1, ^2) and classical input variables (^0, ^1 , ^2 , ^3 , ^4 , ^5). It is noted that the classical input variables have to be mapped to angles, with congruence modulo 2π. The easiest way of doing this for a particular real-valued dataset is dividing all numbers in the dataset by the largest member of the set and then multiplying them all by 2π, meaning that all the amplitude values in the dataset are rescaled to lie in the interval (0, 2^). For datasets with a considerable size variation, alternative methods for mapping the data to the interval (0, 2^), which are per se known from the literature, may be more suitable. Ansatz for F12 Convolutional Filter

[0033] Figure 2 shows an ansatz F12 for a quantum computing device 200 for use as a convolutional filter. The quantum computing device 200 acts on a zeroth, first and second qubit q0, q1, q2, and it comprises: a first rotation block 210 (between lines A and B), which is configured to perform one or more phase rotations on the zeroth qubit conditional on the first qubit; an entangling block 220 (between lines Band C), which is arranged immediately downstream of the first rotation block and configured to provide entanglement between the first and second qubits; and a second rotation block 230 (between lines C and D), which is arranged immediately downstream of the entangling block and configured to perform one or more ^^rotations on the zeroth qubit conditional on the first qubit. The lines A, B, C, D do not represent any operation to be performed in quantum computing device 200.

[0034] In some embodiments, the first rotation block 210 comprises parameterized rotation gates trained such that the output phase of the zeroth qubit approaches a zero-state phase |0^. In particular, the first rotation block 210 may be further configured to perform ^^and ^^rotations on the first qubit, i.e., in addition to the phase rotations on the zeroth qubit.

[0035] In some embodiments, the second rotation block 230 comprises parameterized rotation gates trained such that the output amplitude of the zeroth qubit approaches a zero-state amplitude|0^. In particular, it may comprise a subcircuit configured to prepare a Bell state and at least two parameterized ^^rotation gates.

[0036] In some embodiments, the first and second rotation blocks 210, 230 comprise parameterized rotation gates trained such that, at the output of the quantum computing device, the state of the zero qubit approaches a zero state, that is, |^0^ ≈ |0^.

[0037] In some embodiments, the first rotation block 210 or the second rotation block 230, or both, are unitary. By a known result, this preserves the information in the initial phase.

[0038] The operation of the F12 convolutional filter may be summarized along the following lines. The first rotation block 210 can be trained to transform the phase of q0 in the training data ket vectors (vectors with training data classical input dense angle-encoded) to 0, while preserving the information stored in that phase since the block is unitary. Then the entangling block 220 entangles q1 and q2, and it performs conditional actions between them, in a parametrized way (to provide degrees of freedom for the training to optimize how the entangling works). Finally, the second rotation block 230 can be trained to change the amplitude of q0 in the training data to being 100% in the |0^ state, while preserving the information stored in thatamplitude since the block is unitary. Thus, if training is successful, q0 is already in (or substantially in) the state |0^ when the reset of it (to |0^) in the latent space occurs, and all the information initially in q0 is retained in (or substantially retained in) the entangled state with q1 and q2.

[0039] Figure 3A shows an example implementation, as a structure of parameterized and non-parametrized quantum logic gates, of the quantum computing device 200 introduced in figure 2. In figure 3A and the remainder of the present disclosure, the notation ^0, ^1, ^2, … refer to parameters of the quantum logic gates; this is unlike figure 1, where similarly positioned notation ^0, ^1, ^2, … was used for the classical input variables. The vertical lines figure 3A, which correspond to lines B and C in figure 2, do not represent any operation to be performed, but merely highlight the block structure of the quantum computing device 200.

[0040] In figure 3A, the first rotation block 210 comprises the following subblocks in uninterrupted sequence: an ^^rotation on the zeroth qubit and an ^^rotation on the first qubit; a NOT operation on the zeroth qubit conditioned on the first qubit; an ^^rotation on the zeroth qubit and an ^^rotation on the first qubit; a NOT operation on the first qubit conditioned on the zeroth qubit; an ^^rotation on the first qubit; a NOT operation on the zeroth qubit conditioned on the first qubit; and an ^^rotation on the zeroth qubit and an ^^rotation on the first qubit. The entangling block 220 comprises the following subblocks in uninterrupted sequence: a NOT operation on the first qubit conditioned on the second qubit; a Hadamard gate H on the first qubit followed by an ^^rotation on the first qubit, as well as an ^^rotation on the second qubit; and a NOT operation on the second qubit conditioned on the first qubit. The rotation block 230 comprises the following subblocks in uninterrupted sequence: an ^^rotation on the zeroth qubit and an ^^rotation on the first qubit; a NOT operation on the zeroth qubit conditioned on the first qubit; an ^^rotation on the first qubit; a NOT operation on the first qubit conditioned on the zeroth qubit; an ^^rotation on the first qubit; a NOT operation on the zeroth qubit conditioned on the first qubit; and an ^^rotation on the first qubit.

[0041] It is seen that the depth of the first rotation block 210 is 7, the depth of the entangling block 220 is 4, and the depth of the second rotation block 230 is 7. Hence, the F12 convolutional filter is implemented by a mere 23 gates and has a depth of 18 gates while compressing by 3-to-2. This is significantly shorter than using a 3-to-2filter made from generic two-qubit unitary circuits (each containing 18 gates and being 11 gates deep)

[0023] plus an entangling block (totaling 40 gates, 26 gates deep) or one general three-qubit unitary circuit (contains 138 gates)

[0024] .

[0042] Figure 3B shows an inverse quantum computing device (inverse device) 300 of the quantum computing device 200 in figure 3A. The inverse device 300 comprises an inverse 330 of the second rotation block, which is immediately followed by an inverse 320 of the entangling block, which is immediately followed by an inverse 310 of the first rotation block. The inverse quantum computing device 300 is implemented by forming a sequence of quantum logic cates, which is a reverse of the corresponding sequence in the (forward) quantum computing device 200, and the rotation gates are parameterized with additive inverses of their parameter values in the first sequence. For an ℝ-valued parameter, the additive inverse is obtained by a sign change; the action is a reverse rotation. For example, the final quantum logic gate of the (forward) quantum computing device 200 is an ^^rotation parametrized by ^11; it corresponds to the initial quantum logic gate of the inverse quantum computing device 300, which is an ^^rotation parametrized by −^11. The parameterization is configured before training so that, if a training run results in a parameter adjustment to be applied to ^11, this adjustment affects the behavior of the two corresponding quantum logic gates.

[0043] As an alternative to manual calculation, it is noted that the open-source software development kit Qiskit™, among other tools, can be used to determine inverses of quantum circuits. Quantum convolutional autoencoder

[0044] Figure 4 shows a generic quantum autoencoder structure, which has already been described above.

[0045] Figure 5A shows, according to a second aspect of this disclosure, an example implementation of a quantum convolutional autoencoder (QCAE) 400 that includes a sequence of layers of cascaded instances of the quantum computing device 200 (here denoted Conv_filter_F12) of figure 3A and the inverse quantum computing device 300 (here denoted Conv_filter_F12_dg [dagger]) introduced in figure 3B. The QCAE 400 may be said to have a modular architecture.

[0046] More precisely, figure 5A shows an example where the QCAE 400 has ^^= 2 layers of cascaded instances of the quantum computing device 200, and ^^= 2 layers of cascaded instances of the inverse device 300. The QCAE 400 has a com- pression rate of ^^, which is related to the number of layers by ^^= log^ / ^^^. To obtain a desired compression rate of ^∗ ^ , one may use a QCAE with ^log^ / ^^^∗^ layers, the nearest higher integer number. The compression will be slightly stronger, with a rate of

[0047] The first layer of the QCAE has ^ / 3 instances of said quantum computing device 200. In figure 5, ^ = 9. If ^ is not already a multiple of 3, zero-padding by one or two qubits in the |0^ state can be applied.

[0048] For each next layer of the QCAE, the number of instances of said quantum computing device 200 decreases by a factor 2 / 3 by resetting the zeroth output qubit of each quantum computing device in the previous layer.

[0049] As a result of the repeated decreases, the ^^’th layer (innermost layer) of ^ ^ ^^^^ the QCAE 400 will have at most ^ ^ × ^ ^^ ^ instances of said quantum computing device 200. (For some combinations of ^, ^^values, the number of instances may ^ ^ ^^^^ decrease below ^ ^ × ^ ^^ ^ as a result of rounding effects.) In the special case of figure 5, this means that the second layer has = 2 instances of thequantum computing device 200. Because the zeroth output qubit is reset, the overall compression rate will be 4 / 9 as expected.

[0050] The zeroth output qubit are reset no later than in the latent space, i.e., after the ^^’th layer. Immediately downstream of this, the decoder part is added as the (^^+ 1)th through 2^^’th layers of the QCAE 400. The decode part is the inverse of the ^^layers of cascaded instances of the quantum computing device 200.

[0051] In some embodiments, the quantum computing device 200 and hence the QCAE 400 are adapted to accept dense-angle-encoded input values. Specifically, the QCAE 400 may include an initial normalization layer (not shown) which is con-figured to map the dense-angle-encoded input values to a finite real interval, such as (0,2^).

[0052] Table 1 below gives some key numbers for the above architecture of connecting F12 filters including both convolutional encoding and decoding layer, but not the input angle encoding layer (since that is not part of the autoencoder itself) or latent space layer (since the resets there are just a placeholder). More precisely, Table 1 shows a number of mathematical relations between performance indicators (figures-of-merit) of the QCAE circuit 400. Specifically, it shows how various numbers depend on the number of input qubits ^^and number of layers ^^, and in turn how ^ and ^ depend on the desired c ∗ ^^ompression rate ^^and classical input size ^ℝ ^^. Note that the factors 2 in the equations for ^, ^^and ^^^^reflect the fact that the decoder part and the encoder part contribute equally to those quantities. Table 1 Number ^^of gates as function of ^^input qubits and ^^convolutional layers (not including latent space resets). Note that ^^is divisible by 3 when using this architecture, which can be achieved by zero-padding. Circuit depth ^ for ^^convolutional layers Compression rate ^^for ^^convolutional layers Number of layers ^^as a function of the desired compression rate ^∗ ^ Number of input qubits ^^as a function of the number of classical input variables (continuous real variables) ^ℝ ^^Maximum number ^^^^of F12 filters needed as function of NQinput qubits and NL convolutional layers, in autoencoder. (Note that for certain NQand NL it is possible to further reduce ^^^^.) Maximum total number of parameters ^^that need to be trained for ^^input qubits and ^^convolutional layers. (Note that for certain ^^and ^^it is possible to further reduce ^^.) Maximum number of F12 filters needed ^ ^^^ ^ (^) =^2 ^ × ^ ^ ^ in the ^th layer of a circuit utilizing ^^^^^3 3 input qubits. (Note that for certain ^^and ^^it is possible to further reduce ^^^^.)

[0053] Compared to other QAEs with a similar approach of highly general filters based upon general 2 or 3 qubit unitaries, and with a similar compression rate per layer, the values of the depth ^ per ^^layers and number of gates ^^per number of qubits ^^and layers ^^are very competitive; cf.

[0011] ,

[0021] .

[0054] Compared to other QCAEs that also use a several smaller filters in each convolutional layer, this approach proposed herein has the advantage of a reduced number of filters per layer, as they are not overlapping and there is only one layer of filters in each convolutional layer (e.g.

[0021] ) Also, in those QAEs where a single filter / unitary covering the entire qubit register, the depth of that filter tends to increase (polynomially) in the number of qubits

[0013] ,

[0025] ,

[0026] . In the proposed architecture, as seen in Table 1, the depth ^ per convolutional layer is instead independent of the number of qubits.

[0055] Table 1 also shows the maximum number of filters needed in the ^th layer when using ^^qubits. For certain ^^and ^^, however, it is possible to have a ^^lower than the tabulated value.

[0056] To illustrate, figure 5B shows an example implementation of the encoder part of a QCAE with ^ = 12 input qubits and a desired compression rate of 3 / 12. It is imagined that a corresponding decoder part (inverse of the encoder part) is arranged immediately downstream, to form a complete QCAE. For the purpose of achieving a compression rate of at least 3 / 12, the number of convolutional layers is set toand the layers respectively have 4, 2, 2 and 1 instances of the quantum computing device (F12 filter). It is understood that the zeroth output qubit of each such instance is reset, so that there are three latent-space qubits. In figure 5B, these are q7, q10 and q11.

[0057] As mentioned initially, QCAEs of the type described in this section can be put to use in digital communication networks, especially wireless communication systems. Example uses include the following: - The QCAE may be utilized for coding channel state information which is exchanged between a user equipment and a base station. In particular, the learning of a sparse feedback format may be combined with the task of performing channel reconstruction at the base station using an autoencoder neural network. - Spectrum sensing, where the QCAE learns features from RSSI measurements, and a subsequent classifier determines whether the spectrum of input samples is occupied by a primary user or a secondary user. - Dimensionality reduction in the network; the reconstruction error can be utilized to classify anomalies indicating network intrusion. - Detection of anomalies on network performance datasets. - Modeling uncertainty in network flows by capturing a prior distribution.- Nonlinear feature extraction applied to customer data collected from telecom operators, ultimately to enhance the performance of models that predict loss of customers. - Extraction of latent representations that can improve the detection of telecom fraud. Accordingly, the present disclosure provides a user equipment, a base station (network node) and / or a core network component in a wireless communication system which each includes a QCAE configured to perform one or more of the above tasks.

[0058] The data ^0, ^1, … to be represented by dense-angle encoding may in particular be channel state information (CSI) data, numeric features in telecommunication network customer information data (see [5]), key performance indicators for telecommunication networks (see [6]). The proposed QCAEs can be used for the transmission of these types of data. Training of the QCAE

[0059] Figure 6 is a flowchart of a method 600 of training a QCAE. The method may be performed automatically by one or more programmable processors, and optionally with supervision or assistance from a human operator.

[0060] More precisely, the method 600 may be performed on a computing system 1300 which is implemented using hardware that comprises processing circuitry 1301 and memory circuitry 1302. The processing circuitry is communicatively coupled to the memory circuitry, e.g., via one or more buses. The processing circuitry may comprise one or more microprocessors, microcontrollers, hardware circuits, discrete logic circuits, hardware registers, digital signal processors (DSPs), field- programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), or a combination thereof.

[0061] In some embodiments, the processing circuitry comprises a first processing circuit and a second processing circuit that are capable of executing functions in parallel and / or in series. For example, the processing circuitry may comprise classical processing circuitry and / or quantum processing circuitry. In some such embodiments, one or more particular functions are performed on the classical processing circuitry, whereas one or more other functions are performed on thequantum processing circuitry. Accordingly, particular embodiments may take advantage of the classical and quantum processing capabilities of the computing system as is appropriate for the particular computing system provided. According to one particular example, a quantum convolutional network and qubit measurement may be implemented on the quantum processing circuitry, whereas calculation of the cost and / or gradient may be performed on the classical processing circuitry. Other embodiments include other balancing of the execution of particular tasks on the quantum processing circuitry and classical processing circuitry, as appropriate.

[0062] The processing circuitry may be programmable hardware capable of executing software instructions stored, e.g., as a machine-readable computer program in the memory circuitry. The memory circuitry may comprise any non- transitory machine-readable media known in the art or that may be developed, whether volatile or non-volatile, including but not limited to solid state media (e.g., SRAM, DRAM, DDRAM, ROM, PROM, EPROM, flash memory, solid state drive, etc.), removable storage devices (e.g., Secure Digital (SD) card, miniSD card, microSD card, memory stick, thumb-drive, USB flash drive, ROM cartridge, Universal Media Disc), fixed drive (e.g., magnetic hard disk drive), or the like, wholly or in any combination. The memory may comprise computer readable instructions 1303 or a computer program product 1304 comprising computer readable instructions, the computer readable instructions such that when executed by the processor, the computing system executes a method according to an embodiment of the method 600.

[0063] According to particular embodiments of this hardware, the processing circuitry is configured to perform the method 600 to be described, wherein the initial steps 610, 620, 630 may be primarily allocated to the quantum processing circuitry and the optimization in step 640 may be allocated to the classical processing circuitry.

[0064] In the method 600, in an initial step 610, a QCAE according of the type depicted in figure 5 is provided. The QCAE may be defined by the entity performing the method 600 or obtained from a different entity. It is understood that the values of the parameters ^0, ^1, ^2, … of the quantum computing devices have not been assigned values, or just provisional (placeholder) values; the parameter values to be used in productive operation will be determined by the training.

[0065] Further, in a second step 620, a cost function ^^is defined. The cost function ^^may be defined on the basis of a difference between, on the one hand, a z- spin measurement on all zeroth output qubits that are being reset in the latent space and, on the other hand, a label corresponding to an outcome where all z-spin measurements are applied to a zero state |0^. The cost function ^^may be written:denotes a restriction of the z-spin measurement operator ^^^that acts only on ^^. The cost in each training step is calculated as an expectation value of the cost observable C^on encoder output, say, over 1000 runs, for a particular set of variational parameters^θ :where ^^^^(^) represents a circuit performing dense-angle encoding of the classical input vector ^ (training data vector) to create the autoencoder’s input ket vectorrepresents the entire encoder part of the Quantum Convolutional Autoencoder. This ^^^^^^^^^ is constructed by connecting F12 filters in series.

[0066] Returning to the above example autoencoder acting on 9 qubits with a compression rate of 4 / 9, the operator ^^^^^^^^^ in the cost observable can be expressed aswhere the operatordenotes an F12 filter acting on ^^, ^^, ^^(in this order) and identity acting on all further qubits on in the register. The dependence on the parameter vector ^ is implicit. Further, in the case of training a single F12 filter,and where the only qubit to be reset is ^^, the cost function operator simplifies into

[0067] In a third step 630 of the method 600, a set of DAE training vectors is obtained, such that each training vector represents a quantum state. Any set of training vectors can be used that is considered representative of the intended use case of the QCAE.

[0068] In some embodiments, to reduce the total training load, the set of training vectors consists of substantially orthogonal (orthogonal within numerical approximation) vectors. As the disclosure demonstrates and as will be further discussed below, a collection of ^ orthogonal vectors may be sufficient to perform training in a Hilbert space of dimension ^. A comparable easing of the training load is obtained even for non-perfect orthogonality, e.g., if the set of training vectors contains a small number of vectors in addition to the optimal number ^.

[0069] Then, in a fourth step 640, the parameter values in the QCAE are optimized using the set of training vectors and with respect to the cost function. This is to say, the parameter values are varied (within their predefined ranges) to handle the training vectors optimally, as gauged by the cost function. The optimization 640 may use a gradient-descent algorithm, preferably gradient descent with momentum.

[0070] In some embodiments, the optimization step 640 stops when the set of training vectors has been used up. Alternatively, an assessment of the performance of the QCAE can be performed (e.g., by applying it to a validation set or by considering the cost-function values observed during the optimization 640), which could trigger an earlier end of the optimization step 640 or, as need, an additional execution period.

[0071] Once optimized parameter values ^∗have been found by the training method 600, the reproduced ket vector |Ψ^^^ output by the complete autoencoder is formally described bywhere the product in the central bracket represents the action of the resets in the latent-space layer. For this point, reference is made to the above description of the modular QCAE architecture.

[0072] F12 filters for a single-layer architecture could be trained to reconstruct training data with a reconstruction infidelity consistently around or below 1% when two orthogonal training vectors for each F12 filter (equivalently, 2^ / 3 training vectors) were used. When three orthogonal training vectors per F12 filter were used, it resulted in a more strongly varying reconstruction infidelity usually around 10%. It is noted that, according to the conclusions from the experiments to be discussed below, any vectors that lie in the space spanned by the training vectors can be reconstructed with the same reconstruction infidelity as the ones for the training vectors. The experiments were performed with a few hundreds of training vectors and a few hundreds of testing vectors, which were different from the training vectors.

[0073] Training a connected architecture with more than one convolutional layer is relatively more challenging. The risk of having the optimization converge to a local minimum, where the error on one of the training datapoints is minimized while the error on the other training datapoints is high (above 90%), should preferably be controlled. To avoid such undesirable local minimization, the cost function ^^may be modified such that higher-order powers of the expectation value of the error on each datapoint are summed. This way, such behaviors where the distance from the zero state |0^on the qubits far away from |0^ is decreased are rewarded more than decreasing this distance on qubits close to |0^.

[0074] When testing the QCAE, one can use the fidelity between reconstructed state and input state. Alternatively, one may consider the expectation value of the z- measurements (if using angle encoding) and compare those from the reconstructed state with those from the input state. Then, a measure such as mean square error (MSE) can be calculated over those results from all the testing datapoints. Approaches of this nature are used in the next sections. Trials 1–3 and results

[0075] Experiments were conducted to evaluate the proposed QCAE 400. The following hypothesis was formulated and assessed: Hypothesis: In a quantum autoencoder with non-unitary reset, the infidelity of reconstructed state corresponding to a test vector spanned by the training vectors, is bounded by the worst reconstruction infidelity of state corresponding to any individual training data vector.

[0076] To test the hypothesis, the F12 convolutional filter quantum circuit, trained on orthogonal dense angle encoded training vectors (two at a time, per filter) is used. The orthogonality of these training vectors refers to having the inner product of the ket-vectors after dense angle encoding of the two training datapoints A and B be zero. I.e., ifis the ket after dense-angle encoding the classical input vector of datapoint A, then:To achieve this orthogonality, the first qubit was fed with (^, 0) in datapoint A and (^, 2^) in datapoint B. The data for each of the three trials is shown in Table 2. Table 2 Datapoint A3.141592653589793 3.141592653589793 3.1415926535897930 0 0 5.359196311257092 2.8878983397803513 1.886067254061676 3.388406075717787 0.5230711172193331 4.147603905059486 5.089862619270585 4.649372636013614 1.6638996406737983 5.09657660491925 3.9965125355663567 2.8549664659003935 Datapoint B3.141592653589793 3.141592653589793 3.1415926535897936.283185307179586 6.283185307179586 6.283185307179586 2.6149280762618616 6.0595862160897225 5.254779256454794 1.278665525663155 3.0533244544784313 0.4982189053465453 0.6394667917439397 0.5092995317187441 0.31160251710009695 5.961189421807812 4.151574483743011 3.7773738161626

[0077] As seen in Table 2, the hypothesis testing was based on three training data sets, denoted Trials 1, 2 and 3. Each trial contains two datapoints. The datapoints are randomly generated in the interval (0, 2π), and then the first two elements aremanually set to (π, 0) in datapoint A and (π, 2π) in datapoint B, to ensure orthogonality of the ket vectors corresponding to the two datapoints.

[0078] Different values of certain hyperparameters, including the learning rate and momentum constant of the gradient-descent algorithm (beta), were used in the three trials. The resulting maximum reconstruction infidelity values are reported in Table 3. Table 3 Iteration Parameters Cost Max Function Reconstruction Infidelity Trial 1 1Learning rate=0.01,0.0001869 0.01005780388508 Beta=0.95, Training steps = 120 0.0001869 0.009648355747046 2 Learning rate=0.001, Beta=0.5, Training steps = 120 Trial 2 1Learning rate=0.01,0.3906250 Beta=0.95, Training steps = 80 0.3230324 0.120195601094 2 Learning rate=0.001, Beta=0.1, Training steps = 80 Trial 3 1Learning rate=0.01,0.02887 0.0374633696306 Beta=0.95, Training steps = 120 0.035892 0.03721926 2 Learning rate=0.001, Beta=0.5, Training steps = 80

[0079] In Table 3, the term “Iterations” refers to that the circuit was first trained with one set of hyperparameters for a certain number of steps (Iteration 1). Then, the results from that iteration including final training parameter values are kept, andstarting from those training parameters the circuit is trained with another set of parameters for a certain number of steps (Iteration 2).

[0080] The cost function convergence results are shown in figures 7–12. More precisely, figure 7 is a plot of results obtained in Trial 1, showing cost function as a function of training step for learning rate = 0.01 and beta = 0.95; figure 8 is a plot of results obtained in Trial 1, showing cost function as a function of training step for learning rate = 0.001 and beta = 0.5, using as initial values the parameters learned during the training session plotted in figure 7; figure 9 is a plot of results obtained in Trial 2, showing cost function as a function of training step for learning rate = 0.01 and beta = 0.95; figure 10 is a plot of results obtained in Trial 1, showing cost function as a function of training step for learning rate = 0.001 and beta = 0.1, using as initial values the parameters learned during the training session plotted in figure 9; figure 11 is a plot of results obtained in Trial 3, showing cost function as a function of training step for learning rate = 0.01 and beta = 0.95; and figure 12 is a plot of results obtained in Trial 3, showing cost function as a function of training step for learning rate = 0.001 and beta = 0.5, using as initial values the parameters learned during the training session plotted in figure 11. It is seen that for the training vectors used in Trial 1, the maximum reconstruction infidelity is 0.009648.

[0081] Next, this trained autoencoder is texted on some vectors that are linear superpositions of the two training vectors. This is achieved by creating a superposition between the two training vectors parametrized by a variable a, such thatwhich is then normalized. Table 4 shows reconstruction infidelities for different values of a for the three trials. Table 4 a Reconstruction infidelity Trial 1 Trial 2 Trial 3 0.9 0.008977145 0.120094947 0.03711349407090.8 0.0089975952 0.11996392 0.7 0.0090454490 0.119807796 0.036774743159 0.6 0.0091240179 0.11964889 0.5 0.009227950 0.11952573296 0.036466582867 0.4 0.0093426182 0.1194720169 0.3 0.0094502093 0.119491801953 0.036654917706 0.2 0.0095379159 0.1195615193 0.1 0.0096013900 0.11965206275 0.036974178231

[0082] The data in Table 4 agrees with the Hypothesis formulated above. It shows that, for dense-angle encoded input, with linearly independent (orthogonal) datapoints, if the F12 filter could be trained to learn to reconstruct the training data with a maximal reconstruction infidelity ξ for a particular training dataset, then test data ket vector in the space spanned by the training data ket vectors can also be reconstructed with an error bounded by said infidelity ξ . Trial 4 and results

[0083] A further demonstration of the performance of the F12 autoencoder (see figure 5) on a real data set containing a training set with 17,000 datapoints and test set with 3,000 datapoints (California Housing Dataset

[0027] ) was conducted. The demonstration is referred to as Trial 4.

[0084] This data set contains nine features, which were angle-encoded onto nine qubits. The nine qubits were split into three groups of three each, and each group was autoencoded by an F12 filter acting on three qubits. Since the three F12 filters are independent of each other in this parallel architecture, the F12 filters can be trained and tested independently of each other, on their respective subsets of the input features.

[0085] The parameters of the quantum circuit constituting the F12 autoencoder were trained on subsets (of varying size) of the training data set. As cost function, the infidelity of the trash state with |0^ was used, meaning that the training is penalized if the trash state differs from |0^.

[0086] The fitted model (quantum circuit with learned parameters) is then tested on the 3,000 test datapoints. For each datapoint in the test set, the autoencoder is fed with a quantum state with the datapoint angle encoded onto it, then the state is encoded, compressed and decoded. Finally, the z-spin expectation values of the qubits in the reconstructed state are measured, and they are compared to those of the input angle-encoded state, and the mean square error (MSE) is calculated.

[0087] This process is repeated for all datapoints in the testing set, and the average of the MSE from each datapoint is calculated.

[0088] It was found that the parameters obtained by the training (and by extension, the reconstruction MSE) depended heavily on the initial guess of the parameters. This is a common phenomenon in quantum machine learning, with barren plateaus in parameter space. Often the training would end up in a point in parameter space somewhat close to the initial guess spot.

[0089] In view of this finding, a sweep over a large amount of random initial guesses was conducted, to see which such initialization would yield the best result after training. This sweep suggested that in general, when the COBYLA optimizer is used and training is performed on 100 datapoints from the training set, a MSE around 0.015 to 0.035 (exact value depends largely on initial guess) is obtained. The result appeared to be fairly stable to variations in the number of data, up or down from the original number 100. The best result obtained for this data set was MSE = 0.0145. Closing remarks

[0090] The present disclosure has covered - convolutional filters with an ability to compress dense angle encoded input data (and dense angle encoded intermediate data between convolutional layers); - filters designed on the basis of ansatzes conceived to work well with this compression task (designed to be sufficiently general to perform theirpurposed types of operations on qubits, with some degrees of freedom for optimization, while trying to avoid unnecessary circuit width and depth) and which can be optimized using machine learning methods; - a short circuit-depth quantum convolution auto encoder; - a training method which reduces the total training load by exploiting orthogonality of the encoded quantum states.

[0091] The aspects of the present disclosure have mainly been described above with reference to a few embodiments. However, as is readily appreciated by a person skilled in the art, other embodiments than the ones disclosed above are equally possible within the scope of the disclosure, as defined by the appended patent claims. REFERENCES 1.T. O’Shea and J. Hoydis, "An Introduction to Deep Learning for the Physical Layer," in IEEE Transactions on Cognitive Communications and Networking, vol.3, no.4, pp.563-575, Dec.2017, doi: 10.1109 / TCCN.2017.2758370. 2.S. Dörner, S. Cammerer, J. Hoydis and S. t. Brink, "Deep Learning Based Communication Over the Air," in IEEE Journal of Selected Topics in Signal Processing, vol.12, no.1, pp.132-143, Feb.2018, doi: 10.1109 / JSTSP.2017.2784180. 3. Wen, C.K., Shih, W.T. and Jin, S., 2018. Deep learning for massive MIMO CSI feedback. IEEE Wireless Communications Letters, 7(5), pp.748-751. 4. Subekti, A., Pardede, H.F. and Sustika, R., 2018, November. 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Claims

CLAIMS 1. A quantum computing device (200) for use as a convolutional filter acting on a zeroth, first and second qubit (q0, q1, q2), the quantum computing device comprising: a first rotation block (210), which is configured to perform one or more phase rotations on the zeroth qubit conditional on the first qubit; an entangling block (220), which is arranged immediately downstream of the first rotation block and configured to provide entanglement between the first and second qubits; and a second rotation block (230), which is arranged immediately downstream of the entangling block and configured to perform one or more ^^rotations on the zeroth qubit conditional on the first qubit.

2. The quantum computing device (200) of claim 1, wherein the first rotation block (210) comprises parameterized rotation gates trained such that the output phase of the zeroth qubit approaches a zero-state phase (|0^).

3. The quantum computing device (200) of claim 1 or 2, wherein the second rotation block (230) comprises parameterized rotation gates trained such that the output amplitude of the zeroth qubit approaches a zero-state amplitude (|0^).

4. The quantum computing device (200) of any of the preceding claims, wherein the first and second rotation blocks (210, 230) comprise parameterized rotation gates trained such that, at the output of the quantum computing device, the state of the zeroth qubit approaches a zero state (|^0^ ≈|0^).

5. The quantum computing device (200) of any of the preceding claims, wherein the first and / or the second rotation blocks (210, 230) are unitary.

6. The quantum computing device (200) of any of the preceding claims, wherein the first rotation block (210) is further configured to perform ^^and ^^rotations on the first qubit.

7. The quantum computing device (200) of any of the preceding claims, wherein the first rotation block (210) has a depth of 7.

8. The quantum computing device (200) of any of the preceding claims, wherein the first rotation block (210) comprises the following subblocks in uninterrupted sequence: an ^^rotation on the zeroth qubit and an ^^rotation on the first qubit; a NOT operation on the zeroth qubit conditioned on the first qubit; an ^^rotation on the zeroth qubit and an ^^rotation on the first qubit; a NOT operation on the first qubit conditioned on the zeroth qubit; an ^^rotation on the first qubit; a NOT operation on the zeroth qubit conditioned on the first qubit; and an ^^rotation on the zeroth qubit and an ^^rotation on the first qubit.

9. The quantum computing device (200) of any of the preceding claims, wherein the entangling block (220) has a depth of 4.

10. The quantum computing device (200) of any of the preceding claims, wherein the entangling block (220) comprises a subcircuit configured to prepare a Bell state and at least two parameterized ^^rotation gates.

11. The quantum computing device (200) of any of the preceding claims, wherein the entangling block (220) comprises the following subblocks in uninterrupted sequence: a NOT operation on the first qubit conditioned on the second qubit; a Hadamard gate on the first qubit followed by an ^^rotation on the first qubit, as well as an ^^rotation on the second qubit; and a NOT operation on the second qubit conditioned on the first qubit.

12. The quantum computing device (200) of any of the preceding claims, wherein the second rotation block (230) has a depth of 7.

13. The quantum computing device (200) of any of the preceding claims, wherein the second rotation block (230) comprises the following subblocks in uninterrupted sequence: an ^^rotation on the zeroth qubit and an ^^rotation on the first qubit; a NOT operation on the zeroth qubit conditioned on the first qubit;an ^^rotation on the first qubit; a NOT operation on the first qubit conditioned on the zeroth qubit; an ^^rotation on the first qubit; a NOT operation on the zeroth qubit conditioned on the first qubit; and an ^^rotation on the first qubit.

14. A quantum convolutional autoencoder, QCAE (400) acting on ^ input qubits and having a compression rate of ^^, the QCAE comprising a sequence of: ^^layers of cascaded instances of the quantum computing device (200) of any of the preceding claims; and ^^layers of cascaded instances of an inverse device (300) of said quantum computing device (200), wherein the inverse device comprises an inverse (330) of the second rotation block, which is immediately followed by an inverse (320) of the entangling block, which is immediately followed by an inverse (310) of the first rotation block, wherein: the number of layers is related to the compression rate by ^^= log^ / ^^^; the first layer of the QCAE has ^ / 3 instances of said quantum computing device (200), where ^ is a multiple of 3; for each next layer of the QCAE, the number of instances of said quantum computing device (200) decreases by a factor 2 / 3 by resetting the zeroth output qubit of each quantum computing device in the previous layer; and ^ ^ ^^^^ the ^^’th layer of the QCAE has at most ^ ^ × ^ ^^ ^ instances of said quantum computing device (200).

15. The QCAE (400) of claim 14, wherein the ^^layers of cascaded instances of the inverse device (300) act jointly as an inverse of said ^^layers of cascaded instances of the quantum computing device (200).

16. The QCAE (400) of claim 14 or 15, wherein: the quantum computing device (200) comprises a first sequence of quantum logic gates, including a plurality of parameterized rotation gates; andthe inverse device (300) comprises a second sequence of quantum logic cates, such that the second sequence is a reverse of the first sequence, and the rotation gates are parameterized with additive inverses of their parameter values in the first sequence.

17. The QCAE (400) of any of claims 14 to 16, which is configured to act on dense- angle-encoded input values.

18. The QCAE (400) of claim 17, further comprising an initial normalization layer configured to map the dense-angle-encoded input values to a finite real interval.

19. The QCAE (400) of any of claims 14 to 18, which comprises at most ^^trainable parameters, where20. A method (600) of training a quantum convolutional autoencoder, QCAE, the method comprising: providing (610) a QCAE according to any of claims 14 to 19; defining (620) a cost function (^^) on the basis of a difference between, on the one hand, a z-spin measurement on all zeroth output qubits that are being reset and, on the other hand, a label corresponding to an outcome where all z-spin measurements are applied to a zero state; providing (630) a set of dense-angle-encoding training vectors, each training vector representing a quantum state; and optimizing (640) parameter values in the QCAE using the set of training vectors and with respect to the cost function.

21. The method (600) of claim 20, wherein the set of training vectors consists of substantially orthogonal vectors.

22. The method (600) of claim 20 or 21, wherein said optimizing (640) comprises using a gradient-descent algorithm.

23. The method (600) of claim 22, wherein said optimizing (640) comprises using a gradient-descent algorithm with momentum.

24. A computer program (1303) comprising instructions which, when executed on at least one processor (1301), cause the at least one processor to carry out the method (600) according to any one of claims 20 to 23.

25. A computer readable medium (1304) storing the computer program (1303) of claim 24.

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