The classical convolutional layer is the pivotal layer in the classical CNN model. This layer performs convolutional operations on the classical image to extract features. Specifically, in the convolutional operation, a data block is obtained from the image using a sliding window, and a dot product is calculated between this data block and a weight matrix referred to as the classical convolutional kernel [29]. Unlike the classical convolutional layer, the main idea of a quantum convolutional layer is to extract features from the image by transforming the data block using a quantum convolutional kernel, also known as a quantum filter. The quantum filter consists of both a data encoding circuit and a parameterized quantum circuit.

Data encoding strategy. In our RE-QCNN model, amplitude encoding is adopted as the data encoding strategy. Let { X d } d = 1 D represent the preprocessed D data blocks extracted from the given classical image using a N × N sliding window with a stride of 1, which contain all the information of the image. Through the amplitude encoding strategy, each X d ∈ R N is encoded onto its respective quantum state denoted as |X _d ⟩. Specifically, X d → | X d 〉 = 1 X d ∑ i = 1 N X d i | i 〉 , (1) where X _di represents the i-th component of X _d and X d = ∑ i = 1 N X d i 2 is the 2-norm of X _d . The N-dimensional vector X _d is encoded onto a superposition state |X _d ⟩ with log  N qubits. As a result, amplitude encoding efficiently reduces the number of qubits within our model. However, its implementation is challenging. Specifically, to prepare an arbitrary quantum state, it is necessary to use a quantum circuit whose size scales exponentially with the number of qubits [38,39]. Nevertheless, when dealing with sparse datasets such as the MNIST dataset employed in our numerical experiments, amplitude encoding can be regarded as an efficient strategy. As mentioned in Refs. [50,51], the preparation of an arbitrary sparse quantum state can be accomplished using a quantum circuit whose size scales polynomially with the sparsity and the number of qubits.

Parameterized quantum circuit. We construct the parameterized quantum circuit based on the QAOA circuit that was originally designed to provide approximations for combinatorial optimization problems. The QAOA circuit starts by constructing two Hamiltonians H _B and H _C . Specifically, the Hamiltonian H _B is given by H B = ∑ q = 1 log ⁡ N σ q x , (2) where σ q x is the Pauli X operator acting on the q-th qubit. The Hamiltonian H _C is defined as H C = ∑ q = 1 log ⁡ N − 1 1 2 I − σ q z σ q + 1 z , (3) where σ q z is the Pauli Z operator acting on the q-th qubit. Subsequently, the p-layer QAOA circuit is constructed by iteratively applying e − i β j H B and e − i γ j H C for p rounds, given by U β , γ = ∏ j = 1 p e − i β j H B e − i γ j H C , (4) where parameter vectors β = [β ₁, …, β _p ], and γ = [γ ₁, …, γ _p ]. Figure 3A displays the structure of the QAOA circuit. In more detail, e − i β j H B and e − i γ j H C can be decomposed into single-qubit and two-qubit gates, as illustrated in Figures 3B, C. Specifically, e − i β j H B = ∏ q = 1 log ⁡ N R x 2 β j q , (5) e − i γ j H C = ∏ q = 1 log ⁡ N − 1 e − i γ j 2 I C N O T q , q + 1 R z − γ j q + 1 C N O T q , q + 1 , (6) where R _x (⋅) _q represents Pauli X rotation operator acting on the q-th qubit, and R _z (⋅) _q denotes the Pauli Z rotation operator acting on the q-th qubit. An important characteristic of the QAOA circuit is parameter sharing, which refers to the sharing of the parameter among quantum gates within the same layer. This characteristic of the QAOA circuit reduces the required parameters for constructing the quantum convolutional layer. Moreover, the presence of two-qubit gates in the QAOA circuit provides it with a strong entanglement capability, resulting in a high expressibility of the quantum convolutional layer [48].

Now, the output X _qcl ( β , γ ) of the quantum convolutional layer is a D × D matrix, given by X qcl β , γ = h 1 β , γ … h D β , γ ⋮ … ⋮ h D − 1 D + 1 β , γ … h D β , γ . (7) Here, for d ∈ {1, …, D}, h _d ( β , γ ) denotes the expectation value of a specific observable H, given by h d β , γ = ⟨ X d | U † β , γ H U β , γ | X d ⟩ . (8)

A common form of H is expressed as H = ∑ s = 1 S c s P s , (9) where c s ∈ R , and P s ∈ { I , σ x , σ y , σ z } ⊗ log ⁡ N . Specifically, we can use H = ∑ s = 1 log ⁡ N σ s x , where σ s x represents the Pauli X operator acting on the s-th qubit.

Pooling layer. We employ a downsampling function denoted as down (⋅) to reduce the dimension of X _qcl ( β , γ ), and the resulting output X _pl ( β , γ ) is represented as X p l β , γ = d o w n a X qcl β , γ , (10) where a (⋅) represents an activation function, such as the ReLU activation function. Here, down (⋅) is implemented by computing the maximum of different blocks of a (X _qcl ( β , γ )). Specifically, these blocks can be obtained by applying a 2 × 2 sliding window with a stride of 2 to a (X _qcl ( β , γ )), leading to the dimension of a (X _qcl ( β , γ )) being reduced from D × D to D / 2 × D / 2 .

Fully connected layer. The fully connected layer maps the features extracted by the quantum convolutional layer and the pooling layer to the m-dimensional label space. Specifically, the output y _out ( β , γ , W , b ) of the fully connected layer can be represented as y out β , γ , W , b = g W T X p l β , γ + b , (11) where W denotes the D/4 × m weight matrix, X _pl ( β , γ ) is flattened into a D/4-dimensional vector, b represents the m-dimensional bias vector, and g (⋅) is an activation function, such as the softmax activation function used for the multi-class classification task.

As described in Section 3.1 and 3.2, the forward propagation flow of our RE-QCNN model involves several mappings. Let { X d } d = 1 D be the preprocessed D data blocks. We denote the mapping of the quantum convolutional layer as f qcl : { X d } d = 1 D → X qcl ( β , γ ) , the mapping of the pooling layer as f _pl : X _qcl ( β , γ ) → X _pl ( β , γ ), the mapping of the fully connected layer as f _fcl : X _pl ( β , γ ) → y _out ( β , γ , W , b ), and the mapping of the cost function as l: y _out ( β , γ , W , b ) → C ( β , γ , W , b ). Now, the forward propagation flow is given by l ◦ f fcl ◦ f p l ◦ f qcl . (12)

Here, we focus on the parameter updates of β = [β ₁, …, β _p ] and γ = [γ ₁, …, γ _p ] in the quantum convolutional layer. Following the principles of backpropagation, for j ∈ {1, …, p}, the derivative of C ( β , γ , W , b ) with respect to γ _j is given by ∂ C β , γ , W , b ∂ γ j = ∂ C β , γ , W , b ∂ y out β , γ , W , b ⋅ ∂ y out β , γ , W , b ∂ X p l β , γ ⋅ ∂ X p l β , γ ∂ X qcl β , γ ⋅ ∂ X qcl β , γ ∂ γ j = ∑ d = 1 D ∂ C β , γ , W , b ∂ y out β , γ , W , b ⋅ ∂ y out β , γ , W , b ∂ X p l β , γ ⋅ ∂ X p l β , γ ∂ h d β , γ ⋅ ∂ h d β , γ ∂ γ j . (13)

The first term relies on the specific form of the cost function. Specifically, the cost function quantifies the difference between the predicted labels of our model and the true labels. Different tasks typically use different cost functions. For instance, in the case of a multi-class classification task, cross-entropy is commonly employed as the cost function. Additionally, the second and third terms are associated with the mappings of the fully connected layer and the pooling layer, respectively. These three terms retain classical structures and can be calculated using traditional classical methods. Now, we focus on the calculation of the fourth term. It is crucial to note that the QAOA circuit exhibits parameter sharing, meaning that the same parameter is shared among quantum gates within the same layer. Considering this characteristic, we present the analytical expression for the derivative ∂h _d ( β , γ )/∂γ _j .

Taking the 3-qubit QAOA circuit shown in Figure 4 as an example, we divide this QAOA circuit into five components: V ₁, U ₁ (−γ _j ), V ₂, U ₂ (−γ _j ), and V ₃. Let V 1 , U 1 ( − γ j ) , V 2 , U 2 ( − γ j ) , and V 3 denote the unitary channels corresponding to each component, respectively. Then, the expectation value h _d ( β , γ ) described by Eq. 8 can be reformulated as h d β , γ = t r V 1 ◦ U 1 − γ j ◦ V 2 ◦ U 2 − γ j ◦ V 3 ρ d H , (14) where ρ _d = |X _d ⟩⟨X _d |. Then, we have ∂ h d β , γ ∂ γ j = ∂ t r V 1 ◦ U 1 − γ j ◦ V 2 ◦ U 2 − γ j ◦ V 3 ρ d H ∂ γ j = t r V 1 ◦ ∂ U 1 − γ j ∂ γ j ◦ V 2 ◦ U 2 − γ j ◦ V 3 ρ d H + t r V 1 ◦ U 1 − γ j ◦ V 2 ◦ ∂ U 2 − γ j ∂ γ j ◦ V 3 ρ d H . (15)

Based on the parameter shift rule (see Supplementary Material), for i ∈ {1, 2}, each Pauli Z rotation channel U i ( − γ j ) satisfies ∂ U i − γ j ∂ γ j = U i − γ j + π 2 − U i − γ j − π 2 2 . (16)

Then, we have ∂ h d β , γ ∂ γ j = t r V 1 ◦ U 1 − γ j + π 2 − U 1 − γ j − π 2 2 ◦ V 2 ◦ U 2 − γ j ◦ V 3 ρ d H + t r V 1 ◦ U 1 − γ j ◦ V 2 ◦ U 2 − γ j + π 2 − U 2 − γ j − π 2 2 ◦ V 3 ρ d H = 1 2 t r V 1 ◦ U 1 − γ j + π 2 ◦ V 2 ◦ U 2 − γ j ◦ V 3 ρ d H − 1 2 t r V 1 ◦ U 1 − γ j − π 2 ◦ V 2 ◦ U 2 − γ j ◦ V 3 ρ d H + 1 2 t r V 1 ◦ U 1 − γ j ◦ V 2 ◦ U 2 − γ j + π 2 ◦ V 3 ρ d H − 1 2 t r V 1 ◦ U 1 − γ j ◦ V 2 ◦ U 2 − γ j − π 2 ◦ V 3 ρ d H . (17) Indeed, by estimating the aforementioned four expectation values, we can obtain this derivative. The described process can naturally be extended to the QAOA circuit with log  N qubits. Similarly, we can obtain ∂C ( β , γ , W , b )/∂β _j . Subsequently, by utilizing a gradient-based optimization method, we can update the parameters β and γ in the quantum convolutional layer.

In this section, we provide a detailed analysis of the complexity associated with the forward and backward propagation processes in the quantum convolutional layer.

For the forward propagation process, the D data blocks { X d } d = 1 D are extracted from the given classical image using a N × N sliding window with a stride of 1, which contain all the information of the image. Subsequently, { X d } d = 1 D are sequentially processed by the quantum convolutional kernel, and the resulting D expectation values { h d ( β , γ ) } d = 1 D as described in Eq. 8 form the output of the quantum convolutional layer, represented as a D × D matrix. In detail, the quantum convolutional kernel consists of the data encoding circuit and the parameterized quantum circuit. On one hand, the data encoding circuit adopts the amplitude encoding strategy as described in Eq. 1, and each X d ∈ R N is encoded onto its corresponding quantum state |X _d ⟩ using log  N qubits. Notably, as stated in Theorem 1 of Ref. [50], when X _d is k-sparse (with k nonzero entries), the amplitude encoding strategy is achieved by a data encoding circuit of size O ( k ⁡ log ⁡ N ) , which includes O ( k ⁡ log ⁡ k + log ⁡ N ) single-qubit gates and O ( k ⁡ log ⁡ N ) CNOT gates. Additionally, a quantum superposition query is required to obtain the k nonzero entries of X _d [52]. On the other hand, the parameterized quantum circuit is constructed based on the QAOA circuit U ( β , γ ) with log  N qubits, as described in Eq. 4. According to Eq.5 and 6, e − i β j H B can be decomposed into log  N single-qubit gates, while e − i β j H C can be decomposed into log  N single-qubit gates and 2 log  N CNOT gates in each layer of the QAOA circuit. Therefore, the total number of quantum gates required to implement U ( β , γ ) is O ( p p o l y ( log ⁡ N ) ) , where p is the number of layers. Finally, by performing measurements, we estimate each h _d ( β , γ ) with a complexity of O ( ( k + p ) p o l y ( log ⁡ N ) / ϵ 2 ) , where ϵ represents the additive error in estimating this expectation value. Therefore, the overall complexity of the forward propagation process in the quantum convolutional layer is O ( D ( k + p ) p o l y ( log ⁡ N ) / ϵ 2 ) .

For the backward propagation process, the complexity of the quantum convolutional layer primarily arises from the estimation of { { ∂ h d ( β , γ ) ∂ γ j } j = 1 p } d = 1 D and { { ∂ h d ( β , γ ) ∂ β j } j = 1 p } d = 1 D . As each ∂ h d ( β , γ ) ∂ γ j and ∂ h d ( β , γ ) ∂ β j can be obtained according to Eq. 17, the overall complexity of the backward propagation process in the quantum convolutional layer is O ( D p ( k + p ) p o l y ( log ⁡ N ) / ϵ 2 ) .

Increasing the number of layers p in the parameterized quantum circuit enhances its expressibility, thereby improving the numerical performance of our model. However, when the parameterized quantum circuit reaches a specific number of layers, it may exhibit the barren plateau phenomenon [53,54]. This phenomenon poses challenges to the model training and impacts its numerical performance. Taking these factors into account, choosing p to be O ( log ⁡ N ) is reasonable. In addition, the additive error ϵ is considered to be a constant, which is a general choice in the context of quantum machine learning [55]. In this context, the complexity of both the forward and backward propagation processes is O ( D k p o l y ( log ⁡ N ) ) . Now, we can conclude that the complexity of the quantum convolutional layer is O ( D k p o l y ( log ⁡ N ) ) .

Meanwhile, for the forward propagation process of the classical convolutional layer, the aforementioned D data blocks { X d } d = 1 D , which are all k-sparse N-dimensional vectors, are sequentially processed by the classical convolutional kernel. This results in D feature values { h d ( w ) } d = 1 D . Specifically, each h _d ( w ) is given by h _d ( w ) = w ^T X _d , where w is the N-dimensional weight vector [29]. Considering that each X _d is k-sparse, the calculation of each h _d ( w ) has a complexity of O ( k ) . Additionally, N classical queries are required to obtain the k nonzero entries of X _d . Therefore, the overall complexity of the forward propagation process in the classical convolutional layer is O ( D k ) [47]. For the backward propagation process, the complexity of the classical convolutional layer primarily arises from the calculation of { { ∂ h d ( w ) ∂ w j } j = 1 N } d = 1 D . Since each ∂ h d ( w ) ∂ w j is calculated with a complexity of O ( k ) , the overall complexity of the backward propagation process is O ( D N k ) . Now, we can conclude that the complexity of the classical convolutional layer is O ( D N k ) .

The complexity analysis of the quantum and classical convolutional layers, respectively handling D k-sparse N-dimensional data blocks, is shown in Table 1. There is a trade-off between the sizes of D and N, that is D and N form an inverse proportional relationship. As a result, when N is relatively large and the sparsity k of the data block is O ( log ⁡ N ) , the quantum convolutional layer demonstrates a significant reduction in computational resources compared to the classical CNN model.

The MNIST dataset consists of a training set with 60,000 images and a test set with 10,000 images. Each image in this dataset is composed of 784 (28 × 28) pixels. The visualization of the MNIST dataset is depicted in Figure 5.

In detail, we conduct numerical experiments using the 2-layer RE-QCNN model, which includes two quantum convolutional layers and two pooling layers. The objective of these numerical experiments is to recognize handwritten digit images across all categories. The model configuration details can be found in Table 2. In this configuration, the first quantum convolutional layer consists of a single quantum filter implemented using a 5-qubit QAOA circuit with two layers. Specifically, the quantum convolutional layer applies a 5 × 5 sliding window with a stride of 1 to the normalized 28 × 28 image matrix, resulting in 576 data blocks. Each data block, consisting of 25 pixels, is encoded onto its respective quantum state using 5 qubits through the amplitude encoding strategy. Subsequently, the 576 data-encoding quantum states are sequentially processed by the 2-layer QAOA circuit. Finally, the measurement outcomes of the observable H = ∑ s = 1 5 σ s x are obtained, resulting in 576 expectation values. These expectation values form the output of the first quantum convolutional layer, represented as a 24 × 24 matrix. The second quantum convolutional layer comprises 6 quantum filters, and each quantum filter is also implemented using a 5-qubit QAOA circuit with two layers. Before the output of each quantum convolutional layer enters the pooling layer, it is subjected to a ReLU activation function, followed by a downsampling function. Additionally, for this multi-class classification task, the fully connected layer with the softmax activation function generates the predicted label, and cross-entropy is employed as the cost function. We conduct cross-validation experiments using 2-fold, 3-fold, 5-fold, and 7-fold validation on a total of 70,000 images.

For each cross-validation experiment, the Receiver Operating Characteristic (ROC) curve is depicted in Figure 6A. The ROC curves reveal that our model trained using 2-fold cross-validation lacks superior generalization capability. Specifically, at the False Positive Rate (FPR) of 2%, our model achieves the True Positive Rate (TPR) of 89%, indicating a convergence phase. Complete convergence is achieved at the FPR of 10%. In contrast, our model trained using 7-fold cross-validation demonstrates superior generalization capability. The ROC curve exhibits noticeable improvement at the FPR of 2%, and complete convergence is achieved at the FPR of 10%. Furthermore, the stability of our model is evaluated using the confusion matrix, as depicted in Figure 6B. Overall, our model attains high accuracy in recognizing handwritten digit images across all categories.

The Fashion-MNIST dataset consists of grayscale images of fashion products, where each image is composed of 28 × 28 pixels. This dataset comprises 70,000 images from 10 categories, with 7,000 images per category. The training set contains 60,000 images, while the test set contains 10,000 images. The Fashion-MNIST dataset is considered to be more complex compared to the conventional MNIST dataset. The visualization of the Fashion-MNIST dataset is depicted in Figure 7. In detail, we conduct numerical experiments to comprehensively assess our model’s performance on the Fashion-MNIST dataset. The objective of these numerical experiments is to recognize images across all categories using the 2-layer RE-QCNN model, which includes two quantum convolutional layers and two pooling layers. Next, we present the performance of our model as obtained from numerical experiments involving the QAOA circuit with different random initial parameters, various numbers of layers, and different levels of measurement errors.

Firstly, for the numerical experiments involving the QAOA circuit with five different sets of random initial parameters, the model configuration is consistent with the configuration used for the MNIST dataset, with some details being different. Specifically, the first quantum convolutional layer consists of 40 quantum filters. Each of these quantum filters is implemented using a 5-qubit QAOA circuit with four layers. The second quantum convolutional layer comprises 80 quantum filters, and each quantum filter is also implemented using a 5-qubit QAOA circuit with four layers. These differences can be found in Table 3. Additionally, since we are also dealing with a multi-class classification task, cross-entropy is employed as the cost function. The accuracy of our model, using the QAOA circuit with five different sets of random initial parameters, is depicted as a function of epoch in Figure 8A. This figure illustrates that the highest achieved accuracy is 92.94%, the lowest accuracy is 92.20%, and the average accuracy is 92.59%.

Subsequently, using the same model configuration and the optimal initial parameters of the QAOA circuit mentioned above, we assess the performance of our model under measurement errors of 0, 0.03, and 0.05. Figure 8B reveals that the performance of our model is quite resilient to such errors. Considering that the number of layers in the parameterized quantum circuit of our model is relatively shallow, gate errors may be effectively mitigated by error mitigation techniques [56-59], which suggests small gate errors may not significantly affect the performance of our model. Therefore, we only assess the performance of our model under different levels of measurement errors.

Finally, using the same model configuration and the optimal initial parameters of the QAOA circuit mentioned above, we vary the number of layers in the QAOA circuit to 2, 3, and 4. The corresponding accuracy of our model is depicted as a function of epoch in Figure 8C. This figure demonstrates that our model achieves superior performance when a larger number of layers are employed in the QAOA circuit. The result aligns with the viewpoint that increasing the number of layers enhances the entanglement capability of the QAOA circuit.

Overall, our model exhibits excellent numerical performance on the more complex Fashion-MNIST dataset compared to the MNIST dataset.