Multiclass classification using quantum convolutional neural networks with hybrid quantum-classical learning

1 Introduction

Quantum computing is now widely considered as a new paradigm for solving computational problems, which are believed to be intractable for classical computing devices [1–5]. The idea behind quantum computing is to use quantum physics phenomena [2], such as superposition and entanglement. Specifically, in the quantum gate-based model, quantum algorithms are implemented as a sequence of logical operations under the qubits (quantum analogs of classical bits), which comprise the corresponding quantum circuits terminated by qubit-selective measurements [3]. Examples of the problems, for whose quantum speedups are expected to be exponential, are prime factorization [4] and simulating quantum systems [5], for example, modelling complex molecules and chemical reactions [6]. The amount of computing power for such applications, however, significantly exceeds the resources of currently available quantum computing devices. For example, factoring RSA-2048 bit key requires 20 million noisy qubits [7], whereas currently available noisy intermediate-scale quantum (NISQ) devices have about 50–100 qubits [8]. Quantum computing can be also considered in the context of data processing [9] and machine learning applications [10], where the required resources for solving practical problems are expected to be not so high. Still the caveats of quantum machine learning are related to the input/output problems [11]: Although quantum algorithms can provide sizable speedups for processing data, they do not provide advantages in reading classical input data. The cost of reading the input then may in some cases dominate over the advantage of quantum algorithms. One may note that various approaches have been suggested, specifically, amplitude encoding [12], but the problem of the conversion of classical data into quantum data in the general case remains open [11].

The use of NISQ devices in the context of the quantum-classical (variational) model has emerged as a leading strategy for their use in the NISQ era [13, 14]. In such a framework, a classical optimizer is used to train a parameterized quantum circuit [13]. This helps to address constraints of the current NISQ devices, specifically, limited numbers of qubits and noise processes limiting circuit depths. An interesting link between the quantum-classical (variational) model and architectures of artificial neural networks opens up prospects for the use of such an approach for machine learning problems [15–22]. The workflow of variational quantum algorithms, where parameters of circuit are iteratively updated (optimized), resembles classical learning procedures [19].

A cornerstone problem of various machine-learning-based approaches is classification, that it why it has been widely considered from the view point of potential speedups using quantum computing. As it has been demonstrated in Refs. [9, 23], kernel-based quantum algorithms may provide efficient solutions for the classification problem. Specifically, the quantum version of the support vector machine [9] can be used as an optimized binary classifier with complexity logarithmic in the size of the vectors and the number of training examples. A distant-based quantum binary classification has been proposed in Ref. [24]. Alternative versions of binary quantum classifiers have been considered in Refs. [25–29] (for a review, see also Ref. [30]). A natural next step is to consider the multiclass classification, which has been addressed recently in Ref. [31] with the demonstration of the performance on the IBMQX quantum computing platform. This method uses single-qubit encoding and amplitude encoding with embedding of data, so the obtained results are of quite high accuracy for the 3-class classification task. Very recently, an approach based on quantum convolutional neural network (QCNN) [32] has been used for binary classification and a method to extend it to the multiclass classification case has been discussed. We also note that some of the proposed quantum machine learning algorithms have been tested in practically relevant settings, for example, analyzing NMR readings [33, 34] with the trapped-ion quantum computer, learning for the classification of lung cancer patients [35] and classifying and ranking DNA to RNA transcription factors [36] using a quantum annealer, weather forecasting [37] on the basis of the superconducting quantum computer, and many others [38].

In this work, we present a quantum multiclass classifier that is based on the QCNN architecture. The developed approach uses traditional convolutional neural networks, in which few fully connected layers are placed after several convolutional layers. The corresponding learning procedure is implemented via TensorFlowQuantum [39] as a hybrid quantum-classical (variational) model, where quantum output results are fed to softmax cost function with subsequent minimization of it via optimization of parameters of quantum circuit. Then we discuss the modification of a quantum perceptron, which enables us to obtain highly accurate results using quantum circuits with a relatively small number of parameters. The obtained results demonstrate successful solving of the classification problem for the 4-classes of MNIST images.

Our paper is organized as follows. In Section 2, we present the general description of the proposed quantum algorithm that is used for multiclass classification. In Section 3, we provide in-detail discussion of the layer of the proposed quantum machine learning algorithm. In Section 4, we demonstrate the results of the implementation of the proposed algorithm for multiclass image classification for hand-written digits from MNIST and clothes images from fashion MNIST datasets. We conclude in Section 5.

2 General scheme

The core concept that we use here is the hybrid (variational or quantum-classical) approach (for a review, see Refs. [13, 14]). This approach uses parametrized (variational) quantum circuits, where the exact parameters of quantum gates within the circuit can be changed. The general structure of our variational circuit is shown in Figure 1. Below we describe the proposed approach for multiclass classification based on the classical-quantum approach.

FIGURE 1

FIGURE 1. General structure of the proposed quantum neural network structure consisting of several steps: Preliminary scanning using n-qubit filters, pooling, and regular layers.

At the first step, we realize an amplitude encoding of input data, in our case, MNIST images. In fact, due to the high cost of this step [11], we generate a set of encoding circuits, and store their parameters and structures in a memory, thus generating a quantum dataset. We consider MNIST images, which are rescaled from 28 by 28 to 16 by 16 pixels, and, thus, 8 qubits are needed. In terms of the corresponding qubit states, encoded images can be expressed as follows:

${\mathrm{\Psi }}_k=\sum _{m=0}^N{C}_m^k|m\rangle ,(1)$

where k is the index of image and |m⟩ is a qubit register of 8 qubits, which encode index m, and N = 255. Coefficients $C_m^k$ are equal to elements of normalized flatten vectors of images. In general, this approach enables us to pack vector of N double-precision numbers into log₂(N) qubits, and, thus, significantly reduce the size of processed data. It should be noted however that existing algorithms for amplitude encoding scale exponentially with N; further study is needed to overcome this problem.

We first employ the amplitude encoding procedure [12], where ancilla qubits are used for one-hot encoding of the class of target images. Preliminary analysis of encoded images is performed with 3 convolutional layers with the sizes of filters, equal to 4, 3, and 2, respectively. Each such layer consists of 2 sublayers that are needed to maintain translational invariance (at least, partially), and all the filters of the same size contain identical trainable parameters as is the case for classical convolutional neural networks (CCNN). We note that for filters with the size of 3 we need a virtual qubit, which is always set to zero; such a method is needed to fit the filter into 8 qubits in the translationally invariant manner. The convolutional layer with pooling is then placed after preliminary layers; at this step the first reduction of the required qubit number is realized.

As in the classical setup, several fully connected layers are added after convolutional layers (9 layers in our case). The further reduction of qubit numbers is realized after regular layers and subsequent pooling are done (in the same way as it is done after convolutional filters).

The final filter is needed for mixing the information from two parts of divided circuit. In the process of learning the output of final filter would contain the codes of classes: |00⟩, |01⟩, |10⟩, and |11⟩. Output cascade contains four Toffoli gates, which activate the corresponding ancilla qubit; at the end of the quantum circuit we have one-hot encoded by ancillas class of image. Measurement results of ancilla qubits are passed to the softmax activation function. The categorial cross-entropy is then used as the cost function. The subsequent calculations of gradients of the cost function with respect to the parameters of gates are done using parameter shift rule New parameters of quantum gates obtained by the gradient descent step. The detailed structure of all layers is described below.

3 Structure of layers

Here we present the detailed description of the layers that are used in our quantum machine learning algorithm.

3.1 Preliminary scanning using n-qubit filters

The structure of 4-qubit filters is presented in Figure 2A. First of all, ${\mathsf{R}}_Y({\mathrm{\Theta }}_1)$, ${\mathsf{R}}_Y({\mathrm{\Theta }}_2)$, ${\mathsf{R}}_Y({\mathrm{\Theta }}_3)$ and ${\mathsf{R}}_Y({\mathrm{\Theta }}_4)$ rotations are added in order to rotate each of the four qubits separately. We propose to use controlled parameterized rotations ${\mathsf{R}}_Y(\mathrm{\Phi })$ for the entanglement, which is an essential new element in the structure of quantum perceptron. We note that in Ref. [31] authors use the standard controlled $\mathsf{X}$ gates for this purpose. Here, as we demonstrate, the parameterized entanglement scheme provides higher accuracy of image classification due to the more flexible learning algorithm.

FIGURE 2

FIGURE 2. Quantum circuits for preliminary scanning: In (A) the 4-qubit filter with 4-qubit entanglement is shown; in (B) the stack of 3-qubit filters with 4-qubit entanglement is presented; in (C) the 2-qubit filters with 4-qubit entanglement is demonstrated.

In classical machine learning, the linear perceptron is passed through a certain non-linear function, which is essential for the learning process. In the quantum case, instead of summations of neurons we use entanglement of qubits. The degree of entanglement is controlled by the parameters Φ, which makes the learning process more flexible, and, thus, the classification procedure may become more accurate. In fact, many classical activation functions like sigmoid or tanh behave akin to switches, so their values change from 0 or from −1 to 1 in a certain region. In the quantum domain, we can switch from separable (nonentangled) to entangled states, that play the role of non-linearities in classical learning. So far, individual rotations, which are followed by the parameterized entanglement, can be considered as an analog of the perceptron with the non-linearity.

After 4-qubit scanning, smaller-scale filters are applied to analyze obtained quantum feature map in more details. The structure of layers with 3-qubit filters is presented in Figure 2B. In order to rotate 3 individual qubits ${\mathsf{R}}_Y({\mathrm{\Theta }}_1)$, ${\mathsf{R}}_Y({\mathrm{\Theta }}_2)$ and ${\mathsf{R}}_Y({\mathrm{\Theta }}_3)$ gates are added. Similarly to the case of 4-qubit filters, individual rotations are performed by parameterized ${\mathsf{R}}_Y$ gates. We note that even in the case of 3-qubit filters, we use the entanglement of 4 qubits. Even though the entanglement of 3 qubits looks more intuitive in this case, as we show below, the 4-qubit entanglement provides more accurate results on image recognition. More detailed scanning of images is performed by layer with 2-qubit filters; the corresponding circuit is given in Figure 2C (also see Ref. [32]). As in all previous cases, we use 4-qubit entanglement and the filter consists of 4 individual rotations with additional entanglement by CNOT gates. The idea of using 4-qubit entanglement is inspired by classical CNN, where generation of new feature maps is done by summation of the shared weights of previous feature maps and subsequent application of non-linearity.

3.2 Quantum convolutional neural network layer with pooling

After the preliminary scanning step, the obtained quantum state of 8 qubits contains encoding of feature maps. The role of the next layer (see Figure 3) is to analyze these maps in more detail and pick up the most important of them. The scheme of the layer is given in Figure 3B, where the convolutional filter is the same as in Figure 3A. We note that in the pooling circuit, controlled ${\mathsf{R}}_Z$ rotation is activated if the first qubit is at state 1, while the controlled ${\mathsf{R}}_X$ gate is used when the upper qubit is at state 0. This is conceptually similar to the structure proposed in Ref. [32].

FIGURE 3

FIGURE 3. In (A) the convolutional layer with pooling is shown. In (B) the structure of regular layers is illustrated.

3.3 Regular layers

Similarly to the CCNN case, several regular layers are placed after convolutional layers. In our case we add 8 layers, as shown in Figure 3A. In order to get more accurate results, the double entanglement is added after individual rotations. The second reduction of qubit number in the circuit is done by two pooling procedures as in the case of convolution layers. In order to obtain the required structure, we add a final filter at the end of the quantum circuit. As it is shown below, the use of the final filter is essential for obtaining more accurate results of image classifications.

3.4 Toffoli and controlled rotation gates

The practical realization of high-fidelity two-qubit operations on quantum hardware is still a challenging task. The situation is typically more difficult for three-qubit gates, such as a Toffoli gate. Thus, it is necessary to decompose these gates via single- and two-qubit gates, which can be practically performed. The general algorithm of n-controlled rotations is presented in Ref. [40] and for the case of single-controlled rotation it can be expressed as it is shown in Figure 4A. In order to implement the Toffoli gate, we consider the qubit inversion as a rotation operation around X or Y axes and in our case a doubly-controlled ${\mathsf{R}}_Y(\mathrm{\Theta })$ gate is used with the value of Θ = 2π. The circuit is presented in Figure 4B and it corresponds to the representation of the sum of parameterized n-controlled rotations, which are considered in Ref. [40]. A Toffoli gate, in fact, can be considered as a sum of such rotations with n = 2, where Θ angles of all rotations, except the one that is controlled by the 11th combination, are set to zero. The definition of α angles is realized along the lines of the previously described procedure [40]; they are obtained from Θ angles by simple matrix transformation.

FIGURE 4

FIGURE 4. (A) Single-controlled rotation gates in terms of rotations and $\mathsf{C}\mathsf{N}\mathsf{O}\mathsf{T}$ gates. (B) Decomposition of the Toffoli gate in terms of ${\mathsf{R}}_Y$ rotations and $\mathsf{C}\mathsf{N}\mathsf{O}\mathsf{T}$ gates.

We note that multiqubit gate decomposition can be further improved using qudits, which are multilevel quantum systems. As it has been shown, the upper levels of qudits can be used instead of ancilla qubits in the decomposition [41–45].

4 Classification results

We benchmark the proposed quantum machine learning algorithm with the use of hand-written digits from MNIST and clothes images from fashion MNIST datasets. Examples are presented in Figure 5.

FIGURE 5

FIGURE 5. Examples of MNIST digits (top) and MNIST fashion (buttom) images used in experiments.

All the simulations are performed using Cirq python library for the constructions of quantum circuits; TensorFlowQuantum library [39] is used for the implementation of the machine learning algorithm with parameterized quantum circuits. We use the Adam version of gradient descent, with a learning rate equal to 0.00005, and the overall number of trainable parameters in the QCNN circuit is equal to 149. As a metric for the model performance, we simply use the accuracy of the recognition and for more detailed analysis two sets of experiments are done. In all the conducted experiments, parameterized quantum circuits are trained during 50 epochs.

Within the first set, training and classification are completed for the case, when the dataset consists of images which have certain similarities; thus, the classification problem becomes more difficult. We use MNIST images of digits 3, 4, 5, and 6 for this part. Also, fashion MNIST images with labels 0, 1, 2, and 3 are used for this purpose.

The second experimental set is focused on images, which strongly differ from one another thus making the recognition process easier; MNIST images of digits 0, 1, 2, and 3 and fashion MNIST images with labels 1, 2, 8, and 9 are used here. The total number of considered images of each type is given in Table 1.

TABLE 1

TABLE 1. Number of images of each type.

Each image vector is normalized to one since only that type of vector can be used by the amplitude encoding algorithm. The results of image classification are given in Table 2. Quantum circuits for multiclass classification are considered in Ref. [31]. QCNN examples, provided within the documentation of TensorFlowQuantum [39] also can be relatively simply generalized for the case of multiclass classification tasks. In the second column of Table 2 we provide the results of experiments with circuits, similar to previous results [31]. In order to obtain these results we replace all the ${\mathsf{R}}_Y(\mathrm{\Theta })$ used at entanglement steps by CNOT gates. Also, we remove all the parts of the circuit of Figure 1, which are placed after regular layers, i.e., pooling layers, final layers, and the part with Toffoli gates. Entanglement of ancilla qubits with regular layers is done via CNOT gates according to Figure 2 of Ref. [31]. The third column in Table 2 contains the results obtained with the full circuit shown in Figure 1. The significant improvement in the accuracy of the classification results is caused by two facts. Firstly, the usage of parameterized entanglement in our circuit. Secondly, the increased performance may be connected with the fact that our circuit is constructed in a similar way to classical neural networks—we use a qubit reduction procedure analogous to the reduction of the number of layer outputs in the classical case, i.e., reduction until the number of outputs equals the number of target classes. Note that in Figure 1 ancilla qubits are used only at the read-out step and no entanglement is needed between ancillas and other qubits during the computational procedure, significantly simplifying the corresponding quantum hardware requirements. We also compare obtained quantum results with results of the CCNN with a similar number of parameters, which is 188 in our case. The structure of the CCNN is presented in Table 3. Clearly, the classical results are more accurate, indicating that, with a similar number of parameters, the classical model is still more expressive. An analysis of possible quantum advantage in ML tasks is presented in Ref. [46]. In their study, the authors analyze ML models based on kernel functions and show that with enough data provided, classical methods become more powerful than the corresponding quantum algorithms. Thus, additional study is still needed to find ML tasks where quantum algorithms will outperform their classical analogs.

TABLE 2

TABLE 2. Accuracies of classification for quantum and classical convolutional neural networks.

TABLE 3

TABLE 3. Structure of the used classical convolutional neural network with 188 parameters.

Overall, the QCNN can produce accurate multiclass classifications that are qualitatively similar to the classical model if the number of parameters is comparable. A similar level of accuracy has been achieved previously [31] for the 3-class classification problem. Here, we have demonstrated this level of the accuracy for the 4-class classification tasks, which to the best of our knowledge is the first such demonstration.

5 Conclusion

Here we have demonstrated the quantum multiclass classifier, which is based on the QCNN architecture. The main conceptual improvements that we have realized are the new model for quantum perceptron and an optimized structure of the quantum circuit. We have shown the use of the proposed approach for 4-class classification for the case of four MNISTs. As we have presented, the results obtained with the QCNN are comparable with those of CCNN for the case if the number of parameters are comparable. We expect that further optimizations of the perceptron can be studied in the future in order to make this approach more efficient. Moreover, since the scheme requires the use of multiqubit gates, the qudit processors, where multiqubit gate decompositions can be implemented in a more efficient manner, can be of interest for the realization of such algorithms.

Data availability statement

Publicly available datasets were analyzed in this study. This data can be found here: https://deepai.org/dataset/mnist.

Author contributions

DB—formulation of algorithm, AM—software development, AB—software development, DT—project management, AF—project supervision.

Acknowledgments

We thank A. Gircha for useful comments. AM and AF acknowledge the support of the Russian Science Foundation (19-71-10092). This work was also supported by the Priority 2030 program at the National University of Science and Technology “MISIS” under the project K1-2022-027 (analysis of the method) and by the Russian Roadmap on Quantum Computing (testing the MNIST images).

Conflict of interest

Authors DB, AM, AB, and AF were employed at the Russian Quantum Center which provides consulting services. Owing to the employments and consulting activities of authors, the authors have financial interests in the commercial applications of quantum computing.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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