Parallel Structure of Hybrid Quantum-Classical Neural Networks for Image Classification

Parallel Structure of Hybrid Quantum-Classical Neural Networks for Image Classification | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Parallel Structure of Hybrid Quantum-Classical Neural Networks for Image Classification Zuyu Xu, Yuanming Hu, Tao Yang, Pengnian Cai, Kang Shen, Bin Lv, and 5 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4230145/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 23 Jun, 2025 Read the published version in Quantum Information Processing → Version 1 posted 9 You are reading this latest preprint version Abstract Hybrid quantum-classical neural networks (QCNNs) integrate principles from quantum computing principle and classical neural networks, offering a novel computational approach for image classification tasks. However, current QCNNs with sequential structures encounter limitations in accuracy and robustness, especially when dealing with tasks involving numerous classes. In this study, we propose a novel solution - the hybrid Parallel Quantum Classical Neural Network (PQCNN) - for image classification tasks. This architecture seamlessly integrates the parallel processing capabilities of quantum computing with the hierarchical feature extraction abilities of classical neural networks, aiming to overcome the constraints of conventional sequential structures in multi-class classification tasks. Extensive experimentation demonstrates the superiority of PQCNN over traditional concatenative structures in binary classification datasets, displaying heightened accuracy and robustness against noise. Particularly noteworthy is PQCNN's significantly improved accuracy on datasets with 5 and 10 classes. These findings underscore the transformative potential of the PQCNN architecture as an advanced solution for enhancing the performance of quantum-classical-based classifiers, particularly in the domain of image classification. Quantum computing Quantum-Classical networks Parallel Structure Image classification Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1 Introduction The convergence of quantum computing and artificial intelligence (AI) has sparked the emergence of a promising paradigm known as Quantum Neural Networks (QNNs)[ 1 , 2 ]. QNNs represent a fusion of quantum computing principles with the computational framework of artificial neural networks, offering the potential to revolutionize both fields and tackle computationally intractable problems beyond classical techniques [ 3 , 4 ]. The significance of QNNs lies in their capability to address complex optimization tasks across various domains such as optimization, cryptography, and machine learning. By harnessing the inherent parallelism and entanglement of quantum systems, QNNs offer the prospect of achieving exponential speedup in solving complex optimization tasks compared to classical algorithms [ 5 ]. Moreover, QNNs require fewer resources compared to classical AI models and benefit from the advantages of parallel quantum computing architectures [ 6 ]. However, implementing QNNs on current noisy intermediate-scale quantum (NISQ) devices poses significant challenges. Issues such as barren plateau problems and the curse of dimensionality hinder their practical realization [ 7 – 10 ]. To address these challenges, hybrid quantum-classical architectures, termed Hybrid Quantum-Classical Neural Networks (HQCNNs), have emerged as a promising avenue for AI development. These models combine relatively small, realizable quantum circuits with classical neural networks [ 11 – 13 ]. Notably, various studies have extended classical machine learning algorithms into the quantum domain, with examples including quantum support vector machines [ 14 ], quantum nearest-neighbor algorithms [ 15 ], and quantum decision tree classifiers [ 16 , 17 ]. Additionally, inspired by kernel methods in classical machine learning, quantum kernel methods have been proposed, utilizing quantum devices to estimate kernel matrices and offloading subsequent computations to classical computers [ 18 – 20 ]. Despite these advancements, existing serial quantum-classical neural networks (SQCNNs), where information flows sequentially between quantum and classical networks, face limitations that affect network expressivity and trainability [ 21 ]. For instance, SQCNNs are susceptible to input perturbations and exhibit decreased accuracy for multi-class image classification tasks, especially in the presence of noisy input data [ 22 ]. In this study, we address these challenges by introducing a novel Hybrid Parallel Quantum Convolutional Neural Network (PQCNN) for image classification. Figure 1 (a) and Fig. 1 (b) illustrate a comparison between the traditional quantum serial network and our proposed quantum parallel network. By parallelizing quantum layers with classical layers, our aim is to integrate the advantages of quantum and classical computing, thereby constructing a more efficient and powerful network model. Our contributions in this work are as follows: We introduce the concept of Hybrid Parallel Quantum Convolutional Neural Network (PQCNN), which combines quantum and classical computing in a parallel manner, facilitating its application in tasks such as image classification. We detail the structure of the PQCNN model, elucidating the design of quantum and classical layers and their interconnections. Additionally, we explore the impact of multi-layer parallel structures on experimental results. We demonstrate the effectiveness of PQCNN using various datasets and classification scenarios, investigating its robustness by introducing noise. We analyze the effects of different numbers of quantum bits and circuit depths on the experiments. Moreover, we conduct comparative experiments to validate the efficient performance of our model. The remainder of the paper is organized as follows: Chap. 2 delves into Related Work, discussing previous research in hybrid classical-quantum machine learning and quantum transfer learning. Chapter 3 presents our Proposed Model and Quantum Classifier, providing a comprehensive overview of the model's architecture and design. Chapter 4 showcases our Experimental Results, conducting simulations using the Cucumber dataset and the MNIST dataset to validate the effectiveness of our proposed PQCNN model. Finally, in Chap. 5, we draw conclusions, summarize key findings, and reflect on implications and potential future directions in the field of hybrid quantum-classical machine learning. 2 Related works Numerous significant researches have delved into the fusion of quantum computing and machine learning, with the objective of leveraging quantum algorithms to enrich conventional learning approaches. This section provides a succinct overview of the pertinent studies on the quantum computing framework in image classification, along with the ongoing research trajectory and challenges encountered. 2.1 Hybrid Classical-Quantum Machine Learning Hybrid classical-quantum machine learning represents a burgeoning field at the intersection of classical and quantum computing, aiming to combine the strengths of both paradigms to tackle complex computational tasks more efficiently. In this approach, classical machine learning techniques are integrated with quantum algorithms to leverage the computational advantages offered by quantum computing while mitigating the limitations of current quantum hardware. Notable research efforts in this area include the Hybrid Quantum-Classical Neural Network (QCNN) proposed by Liu et al., which combines classical neural networks with variational quantum circuits to enhance learning capabilities [ 23 ]. Similarly, the Quantum Variational Classifier (QVC) introduced by Adhikary et al. (2020) utilizes classical optimization algorithms to train variational quantum circuits for classification tasks, showcasing the potential synergy between classical and quantum methodologies [ 24 , 25 ]. Moreover, hybrid approaches such as Quantum Enhanced Support Vector Machines (QSVM) and Quantum Generative Adversarial Networks (QGAN) have demonstrated promising results in various machine learning tasks [ 26 – 28 ]. Despite the challenges posed by the current limitations of quantum hardware, such as noise and decoherence, hybrid classical-quantum machine learning holds great promise for unlocking new capabilities and achieving breakthroughs in solving complex problems across diverse domains. Ongoing research efforts focus on refining hybrid algorithms, developing novel techniques for integrating classical and quantum components, and exploring applications in areas such as optimization, pattern recognition, and data analysis. As quantum technology continues to advance, hybrid classical-quantum machine learning is poised to play a pivotal role in shaping the future of computational science and artificial intelligence. 2.2 QCNN for image classification with serial structure Quantum Classical Neural Networks (QCNNs) signify a compelling amalgamation of classical and quantum computing principles tailored for image classification endeavors. By harnessing the complementary attributes of classical and quantum elements, these hybrid architectures endeavor to exploit the inherent advantages of each paradigm. Typically, this entails the orchestration of quantum circuits in tandem with classical multi-layer perceptron (MLP) structures. For instance, Mari et al. [ 29 ] elucidated diverse approaches to hybrid transfer learning, amalgamating classical transfer learning techniques with quantum neural networks for image classification tasks. This strategy not only leverages the computational prowess of quantum computers but also integrates methodologies proven successful in classical machine learning, thereby showcasing substantial potential and utility. Furthermore, Ref. [ 30 ] proposed a scale-inspired local feature extraction technique grounded in Quantum Convolutional Neural Networks, demonstrating enhancements in both recognition and classification accuracy. Additionally, Konar et al. introduced a shallow hybrid QCNNs framework aimed at addressing robust image classification challenges amidst noise and adversarial attacks [ 31 ]. However, it's noteworthy that the majority of QCNN structures are inherently serial, a characteristic that significantly amplifies trainability and expressivity [ 32 ], albeit at the cost of heightened susceptibility to overfitting and increased computational complexity. 2.3 PQCNN In classical neural networks, parallel structures have been extensively employed in complex architectures such as Inception [ 33 ], ResNet [ 34 ], and DenseNet [ 35 ]. Unlike the serial approach, parallel structures offer enhanced computational efficiency and expedited training times, effectively capturing features across multiple scales and abstraction levels. These advantages render the parallel structure of neural networks promising for the advancement of machine learning, a potential extension to QCNN. Several researches have shown that the serial architecture of Quantum Convolutional Neural Networks (QCNN) introduces an information bottleneck, directly impeding the network's capacity for representation and impacting its expressive power and performance. As referenced in [ 36 ], Kordzanganeh et al. proposed the Parallel QCNN, which integrates Multi-Layer Perceptron (MLP) and Variational Quantum Circuits (VQCs) in parallel. It has been found that PQCNN can effectively extract both harmonic and non-harmonic features from datasets. Leveraging its unique architecture, PQCNN demonstrates the ability to discern complex patterns and relationships within data, a task that traditional machine learning algorithms may find challenging. This work underscores the potential of parallel structures in QCNN to enhance performance across a broad spectrum of tasks. Currently, there is limited research on parallel network layers in QCNN, and its applicability to image classification tasks remains unexplored, warranting further investigation. 3 Method This paper aims to develop a hybrid quantum-classical classifier tailored for multi-class classification tasks, utilizing a parallel network structure. Specifically, we employ a classical-to-quantum transfer learning approach for image classification, as illustrated in Fig. 1 . Our selected classical network is ResNet18, a renowned Convolutional Neural Network (CNN) model esteemed for its efficacy in classification tasks and pre-trained on the dataset. 3.1 Architecture of PQCNN Our research paper is based on a transfer learning architecture, serving as the foundational framework for our work. Within this framework, we adopt a hybrid parallel classical-quantum circuit, where the final fully connected layer of the base architecture is replaced with a parallel structure for multi-class classification. This parallel combination of classical and quantum components forms the core of our data processing and classification approach, allowing for multi-layer parallelization. The specific framework diagram is depicted in Fig. 2 . 3.2 Quantum Layer In this section, we provide a detailed explanation of the data encoding method, quantum circuit structure, and data decoding process of the quantum-classical multi-class classifier. 3.2.1 Data Encoding Data encoding refers to the process of converting data from one format or representation into another format suitable for transmission, storage, or processing. In the context of quantum computing or quantum information processing systems, data encoding involves converting classical information into quantum states that can be manipulated and processed using quantum algorithms. In this paper, we adopt an angle-based encoding scheme for the quantum-classical parameter mapping, converting classical parameters into quantum parameters and then transforming them back into classical parameters as output [ 37 ]. For a classical input data \(x{=[{x}_{1},\dots ,{x}_{n}]\mathbb{ }\in \mathbb{R}}^{n}\) , We utilize angle encoding to transform the input \(x\) to a quantum state \(\left|x\right.⟩\) , where \(\left|x\right.⟩\) can be written as: $$\left|x\right.⟩= {\otimes }_{i=1}^{N}\text{cos}\left({x}_{i}\right)\left|0\right.⟩+\text{sin}\left({x}_{i}\right)\left|1\right.⟩$$ 1 Angle encoding utilizes N qubits along with a constant-depth quantum circuit for the encoding process. The state preparation unitary, denoted as \({S}_{{x}_{j}}\) , is expressed as the tensor product of individual unitary operators \({U}_{i}\) , where i = 1, 2, …, N. Each \({U}_{i}\) can be written as follows: $${U}_{i}=\left[\begin{array}{cc}\text{c}\text{o}\text{s}\left({x}_{j}^{\left(i\right)}\right)& -\text{s}\text{i}\text{n}\left({x}_{j}^{\left(i\right)}\right)\\ \text{s}\text{i}\text{n}\left({x}_{j}^{\left(i\right)}\right)& -\text{c}\text{o}\text{s}\left({x}_{j}^{\left(i\right)}\right)\end{array}\right]$$ 2 where \({x}_{j}^{\left(i\right)}\) represents the angle associated with the j -th parameter of the classical data being encoded, and \({U}_{i}\) is the corresponding unitary operator acting on the i -th qubit. This formulation enables the encoding of classical parameters into quantum states using angles, facilitating efficient processing within quantum circuits or algorithms. 3.2.2 Variational Quantum Circuits A quantum circuit comprises a sophisticated arrangement of quantum gates, fundamental units for manipulating qubits, akin to classical logic gates. By organizing quantum gates in a specific sequence, intricate circuits are formed to execute diverse quantum algorithms and tasks. Notable quantum gates, including the Hadamard gate, CNOT gate, and phase gate, facilitate qubit interactions, quantum superposition state generation, and manipulation, among other functionalities [ 38 ]. Quantum gates within these circuits are typically unitary, ensuring reversibility for preserving and transmitting quantum information. Our circuit features single-qubit \({\text{R}}_{\text{y}}\left({\theta }\right)\) g gates and two-qubit CNOT gates, constituting a hardware-efficient setup for network training, as depicted in Fig. 2 . The formulas representing the \({\text{R}}_{\text{y}}\left({\theta }\right)\) gate and the \(\text{C}\text{N}\text{O}\text{T}\) gate matrix representation are given as: $$\text{C}\text{N}\text{O}\text{T}=\left[\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}1& 0\\ 0& 1\end{array}\end{array}\right]$$ 3 $${\text{R}}_{\text{y}}\left({\theta }\right)=\left[\begin{array}{cc}\text{c}\text{o}\text{s}\left(\frac{{\theta }}{2}\right)& -\text{s}\text{i}\text{n}\left(\frac{{\theta }}{2}\right)\\ \text{s}\text{i}\text{n}\left(\frac{{\theta }}{2}\right)& \text{c}\text{o}\text{s}\left(\frac{{\theta }}{2}\right)\end{array}\right]$$ 4 The Controlled-NOT (CNOT) gate entangles qubits and incorporates a control parameter, θ, which assists in the training process and accelerates model speed by enabling parallel computation. A variational quantum circuit of depth q is a succession of numerous quantum layers, each representing the product of multiple units parameterized by distinct weight values. This concept can be mathematically expressed as: $${\mathcal{M}={L}_{1}\cup {L}_{2}\cup \bullet \bullet \bullet \cup L}_{\text{q}}$$ 5 where \(\mathcal{M}\) represents the overall structure of the quantum circuit, where L 1 denotes one layer of the quantum circuit as shown in Fig. 2 . The notation \({{L}_{1}\cup {L}_{2}\cup \bullet \bullet \bullet \cup L}_{\text{q}}\) represents the concatenation of q layers of the circuit. The depth of the quantum circuit is also a crucial parameter, representing the number of layers of quantum gates cascaded together. Deeper quantum circuits typically enable the execution of more complex computational tasks, but they also require more qubits and gate operations, necessitating a balance between depth and resource consumption when designing quantum circuits. Quantum circuits serve as the core tools in quantum computing. By strategically designing and optimizing the connections and parameters of quantum gates, various complex quantum computing tasks can be accomplished. 3.2.3 Data Decoding The decoding layer plays a pivotal role in converting the quantum state into classical states, commonly accomplished through measurement operations. During this process, measurement is conducted utilizing the Pauli-Z operator to acquire the expected value of the Z-direction spin component of the qubit, thereby producing the corresponding classical state vector. The specific expression for this operation is as follows: $$\mathcal{F}=\left|X\right.⟩\to Y=⟨X\left|\stackrel{-}{Y}\right|X⟩$$ 6 where \(\left|X\right.⟩\) represents the input quantum state, \(Y\) represents the decoded classical state, and \(\stackrel{-}{Y}\) denotes the measurement operator used for decoding. The objective of this process is to extract classical information from the quantum state, typically represented by the expectation value associated with a specific physical quantity. Hence, the decoding layer assumes a critical role in quantum computing, facilitating the conversion of quantum information into classical information for subsequent processing and analysis. 3.3 Classical Layer In the classical layer segment of the parallel architecture, we implemented two fully connected layers for transition purposes. Each neuron within the fully connected layer establishes complete connectivity with all neurons in its preceding layer. This design allows the fully connected layer to amalgamate localized information containing category-specific discriminative features extracted from convolutional or pooling layers. Furthermore, at the network's culmination, we employed a fully connected layer to serve as the output mechanism for classification, essentially functioning as a sophisticated "classifier" within our comprehensive network framework. 3.4 Loss Function and Evaluation of the PQCNN In the training process of our PQCNN, we employ the cross-entropy loss function to facilitate the classification task. This pivotal metric allows us to measure the dissimilarity between the predicted outcomes generated by our model and the actual ground truth labels associated with the data. The cross-entropy loss function is expressed mathematically as follows: $$Loss=\frac{1}{N}\sum _{i}(-\sum _{j=1}^{M}{y}_{ij}\text{l}\text{o}\text{g}\left({p}_{ij}\right))$$ 6 where M represents the number of categories, \({y}_{ij}\) represents whether the sample i belongs to category j (1 if it does, 0 otherwise), and \({p}_{ij}\) represents the predicted probability that observation i pertains to category j . 4 Experimental Results and Analysis In this section, we delve into the intricacies of our experiments and simulations, showcasing the efficacy of our method across two diverse datasets: the cucumber plant disease dataset and the handwritten digit dataset (MNIST), both utilized for classification tasks. We meticulously examine the results obtained, providing comprehensive insights into the performance and robustness of our approach. 4.1 Experimental Environment and Training Setup Our experimental setup was executed on a local computing system featuring a ten-core CPU clocked at 2.5 GHz. The computational framework employed for our experiments comprised PennyLane and PyTorch. PennyLane, an open-source Python library, enables seamless integration of quantum and classical computing through automatic differentiation. This framework seamlessly interfaces with popular machine learning libraries like TensorFlow and PyTorch. During the training phase, we employed the Adam optimizer with a learning rate of 0.0004 for 15 iterations. The quantum component of our model was composed of 4 qubits, while the circuit depth was set to 6 layers. 4.2 Evaluation of Our Methodology Using Cucumber Plant Disease Dataset Our research focuses on evaluating the effectiveness of our methodology for binary classification employing the cucumber plant disease dataset. This dataset constitutes a vital segment within a comprehensive repository of plant disease data extensively employed in scientific investigations. It encompasses a diverse array of images showcasing different diseases afflicting cucumber plants. The principal purpose of curating this dataset is to expedite the advancement and optimization of machine learning algorithms tailored for automated detection and characterization of plant diseases. To thoroughly examine the performance of our proposed methodology, we conducted experiments involving the introduction of Gaussian noise with different level. Gaussian noise, known for its normal distribution, effectively replicates random disturbances encountered in real-world scenarios [ 39 ]. During the experimentation phase, we manipulated the variance or standard deviation of the Gaussian noise to modulate its intensity. By injecting Gaussian noise at varying levels into the cucumber plant disease dataset, we were able to assess the model's efficacy amidst differing degrees of interference, as shown in Fig. 3 . This enabled a comprehensive evaluation of its performance under diverse environmental conditions. Figure 4 (a) and Fig. 4 (b) illustrate the accuracy and loss metrics extracted from binary classification tasks conducted on the cucumber plant disease dataset, employing a range of network architectures. These visual representations offer valuable insights into the performance dynamics observed across diverse models within the dataset. We conducted experiments utilizing neural networks featuring distinct parallel structures, including parallel quantum-classical (QC), quantum-classical-quantum (QCQ) configurations, among others. Our findings underscore a notable superiority in accuracy for the parallel structure QCN compared to its traditional serial counterpart. For instance, the parallel-QCN achieves an accuracy of 96.1%, contrasting with approximately 94.5% accuracy for the serial QCN. The loss function curves presented in Fig. 4 (b) further elucidate these observations, revealing that the parallel structure's convergence values are consistently smaller than those of the serial structure. These outcomes strongly indicate that parallel structures exhibit superior image recognition capabilities relative to their serial counterparts in the context of QCNs. After introducing Gaussian noise into the data images, the superiority of the hybrid parallel quantum convolutional neural network over its traditional serial counterpart becomes apparent. Despite a marginal decrease in accuracy with escalating noise intensity, the parallel architecture consistently maintains a commendable level of precision. In contrast, the accuracy of the serial configuration exhibits a sharp downward trajectory, as depicted in Fig. 4 (c). This outcome underscores the superior robustness of the hybrid parallel architecture in mitigating noise interference. The parallel structure's efficacy arises from its ability to concurrently process information from multiple branches, thereby offering heightened resistance against the adverse impact of noise on model performance. Conversely, the serial structure is more susceptible to interference in the presence of noise, resulting in a precipitous decline in accuracy. The high accuracy and resilience of PQCNN presumably stem from its streamlined training process leveraging parallel computing and the formidable anti-jamming capability inherent in quantum neural networks as indicated in [ 40 ]. Moreover, the parallelization of multiple quantum or classical layers presents an opportunity to augment the model's capacity and expressive prowess. Envisioning the design of such parallelized structures with multiple quantum or classical layers holds promise for further enhancing the model's robustness and performance, rendering it more dependable and efficient in managing noise and intricate data. Consequently, harnessing the advantages of parallel structures comprehensively during the model's design and optimization stages can furnish us with increased options and adaptability, thereby culminating in superior model performance. 4.3 MNIST Handwritten Digit Dataset for Multi-Class Image Classification In this investigation, we conducted an extensive evaluation of the effectiveness of our proposed methodology within the domain of handwritten digit recognition, using the widely recognized MNIST dataset as our experimental platform. Renowned for its consistency and applicability, the MNIST dataset consists of 70,000 grayscale images, each meticulously labeled to represent handwritten digits spanning from 0 to 9. To streamline our analysis, we meticulously curated a subset from this extensive dataset, ensuring an equitable distribution of 1000 images for each digit category. Subsequently, this subset was divided into distinct training and testing sets, with 800 images allocated for training and the remaining 200 images reserved for rigorous testing. In framing our experimental approach, we delineated two distinct classification tasks: a five-class classification task and a ten-class classification task. For the former, our focus was specifically on digits 3 through 7, resulting in a dataset comprising five thousand images representing this narrower range of handwritten numerals. In contrast, the latter classification task encompassed the entirety of the MNIST dataset, spanning all ten digits (0 through 9) and yielding a comprehensive collection of 10,000 images. To offer visual insights into our experimental configuration, we present representative samples from the 5-classification and 10-classification tasks in Fig. 5 (a) and Fig. 5 (b) respectively. These visualizations provide a glimpse into the diversity and intricacy inherent in the MNIST dataset, reaffirming its pivotal role as a benchmark for assessing the performance of machine learning algorithms in the realm of handwritten digit recognition. In the initial phase of our research, we conducted a thorough examination of the MNIST dataset by employing various parallel multi-layer configurations. We addressed both five-classification and ten-classification tasks to explore the dataset comprehensively. Figure 6 illustrates the performance trajectories, detailing accuracy and loss for each task category. Our analysis revealed a significant performance gap between the QCQCQ model implemented in the parallel multi-layer architecture and its serial counterpart. For the five-classification task, the QCQCQ model exhibited an impressive accuracy of 95.90%, surpassing the accuracy of the serial structure by 17.30%, which stood at 78.6%. This notable improvement underscores the efficacy of the parallel multi-layer architecture. Similarly, in the context of the ten-classification task, the QCQCQ model achieved an accuracy of 93.90%, showcasing a substantial enhancement of 42.62% compared to the serial structure's accuracy of 51.25%. These findings highlight the superiority of the parallel multi-layer configuration in handling complex classification tasks. The elucidation provided by Fig. 7 , showcasing the intricate details of the confusion matrix, offers a profound understanding of the predictive capabilities across individual digits. It meticulously dissects the performance disparities among the hybrid serial CQ, hybrid parallel QC, and hybrid parallel QCQC architectures. Upon scrutinizing the five-class MNIST image classification confusion matrix, as illustrated in Fig. 7 (a), glaring differences emerge. The serial-CQ structure struggles to correctly identify a mere 22 instances out of 200 for the digit “3”, starkly contrasting with the commendable performance of the hybrid parallel QC and QCQC configurations, which accurately classify 179 and 183 instances of the digit "3", respectively. Table 1 The summary of data for binary, five-class, and ten-class classification tasks Dataset Structure Accuracy Precision Recall F1-Score Cucmber-2 Serial-CQ 0.9459 0.9446 0.9462 0.9454 Parallel-QC 0.9639 0.9628 0.9645 0.9636 Parallel-CQC 0.9684 0.9677 0.9685 0.9681 Parallel-QCQC 0.9774 0.9768 0.9777 0.9772 Parallel-QCQCQ 0.9639 0.9623 0.9664 0.9637 Mnist-5 Serial-CQ 0.786 0.7942 0.786 0.7413 Parallel-QC 0.937 0.9369 0.9369 0.9366 Parallel-CQC 0.955 0.9554 0.955 0.9548 Parallel-QCQC 0.955 0.9550 0.9549 0.9549 Parallel-QCQCQ 0.959 0.9590 0.959 0.9588 Mnist-10 Serial-CQ 0.5125 0.4936 0.5125 0.4371 Parallel-QC 0.932 0.9322 0.932 0.9319 Parallel-CQC 0.9365 0.9364 0.9365 0.9362 Parallel-QCQC 0.9375 0.9366 0.9359 0.9359 Parallel-QCQCQ 0.939 0.9398 0.939 0.939 A comprehensive summary of the experimental results is encapsulated in Table 1 . In addition to the accuracy advantages for the parallel-QCN compared with the serial structure, the parallel-QCN also outperforms the serial model across all metrics, including the precision, recall, and F1-score. The superior performance of the parallel-QCN model can be attributed to its ability to capture more information and features present in the images through the quantum and classical layers, as discussed in Ref.[ 36 ]. In contrast, the serial-QCN model, although capable of capturing sequential information, may cause the information bottlenecks, limiting the expressivity of the network. In our exploration of quantum circuit dynamics in relation to classification tasks, we delved into the interplay between the number of qubits and circuit depth, employing the MNIST dataset for a ten-classification task. Initially, with the circuit depth held constant at 6, we conducted experiments varying the number of qubits from 2 to 10. The resultant insights, depicted in Fig. 8 (a), highlighted a notable trend: as the number of qubits increased, experimental accuracy exhibited a progressive enhancement. This phenomenon underscores the pivotal role of qubit quantity in augmenting the quantum circuit's representational prowess. Indeed, by affording greater degrees of freedom, an increased qubit count enables the circuit to adeptly capture the intricate features inherent within the dataset. Conversely, a dearth of qubits may curtail the circuit's capacity to effectively encode complex data patterns. Thus, the observed improvement in accuracy with escalating qubit numbers is emblematic of the circuit's augmented expressive power, facilitating a finer delineation of the dataset's nuanced structure. Subsequently, with the number of qubits set at 4, we scrutinized the impact of varying circuit depth from 2 to 10. The resultant observations, delineated in Fig. 8 (b), revealed an intriguing relationship between depth and classification accuracy. Remarkably, the pinnacle of accuracy was attained at a circuit depth of 4. This finding underscores the delicate balance required in determining an optimal circuit depth. Insufficient depth risks the specter of underfitting, where the circuit fails to capture the intricacies of the dataset, thereby compromising classification accuracy. Conversely, an excessive depth engenders the pernicious encroachment of noise within the circuit []. As the depth amplifies, the concomitant proliferation of qubit interactions may undermine coherence between quantum states, culminating in heightened noise levels detrimental to computational fidelity. Thus, in navigating the labyrinthine terrain of circuit depth optimization, considerations spanning dataset complexity, noise mitigation strategies, and computational resource constraints emerge as pivotal determinants in fostering the zenith of classification performance. In synthesis, our investigation into the nexus of qubit quantity and circuit depth illuminates the nuanced dynamics shaping the efficacy of quantum circuits in classification tasks. By discerning the intricate interplay between these parameters, we endeavor to chart a course towards unlocking the full potential of quantum computational paradigms in the realm of machine learning and beyond. 4.4 Comparative Analysis The comparative analysis conducted in Table 2 highlights the significant advantages offered by our method over alternative quantum approaches in the realm of 5-class and 10-class MNIST image classification. Notably, our method outperforms quantum neural networks referenced in Ref. [ 41 ] and [ 42 ], as well as serial quantum-classical neural networks mentioned in Ref. [ 43 ]. However, it's crucial to acknowledge that the data presented in the table are sourced from various studies, each utilizing distinct dataset specifications. Direct comparisons between methodologies must be approached with caution due to potential discrepancies in dataset composition and characteristics. These variations can obscure the interpretation of results and may lead to misleading conclusions. Therefore, it's essential to view these findings as contextual reference points rather than definitive assessments. While our method demonstrates clear advantages, it's imperative to recognize the complexity of the broader quantum landscape. A nuanced scrutiny of different quantum methodologies is necessary, taking into account the intricacies inherent in dataset variability and experimental design. By doing so, we can gain valuable insights into the relative efficacy of various quantum approaches within their respective experimental contexts, thus advancing the field of quantum image classification. Table 2 Comparison of accuracy with other quantum methods Dataset Accuracy Our method Mnist(3–7) 95.9% Mnist (0–9) 93.9% QNN Ref.[ 41 ] Mnist (0–9) 65.13% QNN Ref.[ 42 ] Mnist (3–6) 85.14% QCNN Ref.[ 43 ] Mnist (0–9) 79.9% 5 Summary In conclusion, our paper introduces the Hybrid Parallel Quantum Classical Neural Network (PQCNN), a novel approach aimed at overcoming challenges encountered in implementing serial Quantum Classical Neural Networks (SQCNN) for image classification tasks. Through a thorough exploration of the PQCNN model, backed by detailed experimental results and analysis, several significant findings have been elucidated. The experimental validation conducted across datasets, including the cucumber plant disease dataset and the MNIST handwritten digit dataset, highlights the effectiveness of PQCNN in achieving high accuracy and robust performance in classification tasks. Analysis of various performance metrics, including accuracy, precision, recall, and F1-score, emphasizes the superior capabilities of PQCNN over SQCNN, particularly in addressing complex classification tasks across diverse datasets. These findings provide valuable insights into optimizing quantum-classical neural network architecture, with considerations ranging from noise mitigation to multiclassification scenarios. Overall, our results contribute to advancing the understanding and application of quantum computing in image classification tasks through the development and validation of the PQCNN model. This research opens avenues for further exploration and investigation in the realm of hybrid quantum-classical machine learning, with potential implications extending beyond image classification to a wide array of applications. Declarations Author Contribution Z. Xu and Y. Hu wrote the main manuscript and conceptualized the study, T. Yang, P. cai, K. Shen, and B. Lv participated in data collection and analysis, S. Chen, J, Wang prepared figure 1-4 and checked the language, Y. Zhu and Z. Wu contributed to the writing and editing of the manuscript, Y. Dai supervised this work. All authors reviewed the manuscript. Acknowledgements This work was partly supported by the National Natural Science Foundation of China (Grant Nos. 61874001, 62004001, 62201005, 62004001, 62304001), the Anhui Provincial Natural Science Foundation under Grant No. 2308085QF213, 2308085QF195, and the Natural Science Research Project of Anhui Educational Committee under Grant No. 2022AH050106, 2023AH050072. 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Astron. 64, 290311 (2021). https://doi.org/10.1007/s11433-021-1734-3 Jäger, J., Krems, R.V.: Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines. Nat Commun. 14, 576 (2023). https://doi.org/10.1038/s41467-023-36144-5 Adhikary, S., Dangwal, S., Bhowmik, D.: Supervised learning with a quantum classifier using multi-level systems. Quantum Inf Process. 19, 89 (2020). https://doi.org/10.1007/s11128-020-2587-9 Bishwas, A.K., Mani, A., Palade, V.: An all-pair quantum SVM approach for big data multiclass classification. Quantum Inf Process. 17, 282 (2018). https://doi.org/10.1007/s11128-018-2046-z Wang, H., Song, Z., Wang, Y., Tian, Y., Ma, H.: Target-generating quantum error correction coding scheme based on generative confrontation network. 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Applied Soft Computing. 136, 110099 (2023). https://doi.org/10.1016/j.asoc.2023.110099 Sim, S., Johnson, P.D., Aspuru‐Guzik, A.: Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum‐Classical Algorithms. Adv Quantum Tech. 2, 1900070 (2019). https://doi.org/10.1002/qute.201900070 Szegedy, C., Wei Liu, Yangqing Jia, Sermanet, P., Reed, S., Anguelov, D., Erhan, D., Vanhoucke, V., Rabinovich, A.: Going deeper with convolutions. In: 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). pp. 1–9. IEEE, Boston, MA, USA (2015) Wang, F., Jiang, M., Qian, C., Yang, S., Li, C., Zhang, H., Wang, X., Tang, X.: Residual Attention Network for Image Classification. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6450–6458. IEEE, Honolulu, HI, USA (2017) Huang, G., Liu, Z., Van Der Maaten, L., Weinberger, K.Q.: Densely Connected Convolutional Networks. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). pp. 2261–2269. IEEE, Honolulu, HI (2017) Kordzanganeh, M., Kosichkina, D., Melnikov, A.: Parallel Hybrid Networks: An Interplay between Quantum and Classical Neural Networks. Intell Comput. 2, 0028 (2023). https://doi.org/10.34133/icomputing.0028 LaRose, R., Coyle, B.: Robust data encodings for quantum classifiers. Phys. Rev. A. 102, 032420 (2020). https://doi.org/10.1103/PhysRevA.102.032420 Mohammadi, M., Niknafs, A., Eshghi, M.: Controlled gates for multi-level quantum computation. Quantum Inf Process. 10, 241–256 (2011). https://doi.org/10.1007/s11128-010-0192-z Rahman, A.U., Noman, M., Javed, M., Luo, M.-X., Ullah, A.: Quantum correlations of tripartite entangled states under Gaussian noise. Quantum Inf Process. 20, 290 (2021). https://doi.org/10.1007/s11128-021-03231-9 Beer, K., Bondarenko, D., Farrelly, T., Osborne, T.J., Salzmann, R., Scheiermann, D., Wolf, R.: Training deep quantum neural networks. Nat Commun. 11, 808 (2020). https://doi.org/10.1038/s41467-020-14454-2 Trochun, Y., Pavlov, E., Stirenko, S., Gordienko, Y.: Impact of Hybrid Neural Network Structure on Performance of Multiclass Classification. In: IEEE EUROCON 2021 - 19th International Conference on Smart Technologies. pp. 152–156. IEEE, Lviv, Ukraine (2021) Bokhan, D., Mastiukova, A.S., Boev, A.S., Trubnikov, D.N., Fedorov, A.K.: Multiclass classification using quantum convolutional neural networks with hybrid quantum-classical learning. Front. Phys. 10, 1069985 (2022). https://doi.org/10.3389/fphy.2022.1069985 Kashyap, S., Garani, S.S.: Quantum Convolutional Neural Network Architecture for Multi-Class Classification. In: 2023 International Joint Conference on Neural Networks (IJCNN). pp. 1–8. IEEE, Gold Coast, Australia (2023) Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 23 Jun, 2025 Read the published version in Quantum Information Processing → Version 1 posted Editorial decision: Revision requested 08 Oct, 2024 Reviews received at journal 07 Jun, 2024 Reviewers agreed at journal 07 Jun, 2024 Reviewers agreed at journal 03 Jun, 2024 Reviewers agreed at journal 29 May, 2024 Reviewers invited by journal 28 May, 2024 Editor assigned by journal 08 Apr, 2024 Submission checks completed at journal 08 Apr, 2024 First submitted to journal 07 Apr, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4230145","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":291927738,"identity":"beb2beed-4668-4053-99cb-415b5f6632c2","order_by":0,"name":"Zuyu Xu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAsklEQVRIiWNgGAWjYLCCDwwHQJQB8ToYZ5CshZmHJC3yM3LMpG1q7iQ2sDdvk2CouUOMo9LSpHOOPUts4DlWJsFw7BkRjpJIPiadw3Y4sUEix0yCseEwYS1sEolt0hb/gFrk3xCphQdkC2MbyBYeIrVI8DxLtuztO2zcxpNWbJFwjAgt8u05hjd+fDss289+eOONDzVEaGEQSIDQbCAigQgNDAz8B4hSNgpGwSgYBSMZAACxkTbUdYlCHAAAAABJRU5ErkJggg==","orcid":"","institution":"Anhui University","correspondingAuthor":true,"prefix":"","firstName":"Zuyu","middleName":"","lastName":"Xu","suffix":""},{"id":291927739,"identity":"8cd80529-de78-43aa-83a9-6c9125ac1df9","order_by":1,"name":"Yuanming Hu","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Yuanming","middleName":"","lastName":"Hu","suffix":""},{"id":291927740,"identity":"0bb225dc-9a89-477c-bfe1-1dd1f0079962","order_by":2,"name":"Tao Yang","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Tao","middleName":"","lastName":"Yang","suffix":""},{"id":291927741,"identity":"16f650ce-04ac-4763-9d08-54c299582ae3","order_by":3,"name":"Pengnian Cai","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Pengnian","middleName":"","lastName":"Cai","suffix":""},{"id":291927742,"identity":"feee433c-52f0-492f-9b90-e7937d37eb92","order_by":4,"name":"Kang Shen","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Kang","middleName":"","lastName":"Shen","suffix":""},{"id":291927743,"identity":"33018c5c-4281-436e-ab60-9a681b10fd7b","order_by":5,"name":"Bin Lv","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Bin","middleName":"","lastName":"Lv","suffix":""},{"id":291927744,"identity":"f8590e15-5ca5-452a-973c-0e7452aa3e3c","order_by":6,"name":"Shixian Chen","email":"","orcid":"","institution":"Anhui Normal University","correspondingAuthor":false,"prefix":"","firstName":"Shixian","middleName":"","lastName":"Chen","suffix":""},{"id":291927745,"identity":"c9647d23-03e8-4ed9-8c74-b96138ce116f","order_by":7,"name":"Jun Wang","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Jun","middleName":"","lastName":"Wang","suffix":""},{"id":291927746,"identity":"171f34cd-ccd9-49d8-8d2b-26501ce2e5c2","order_by":8,"name":"Yunlai Zhu","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Yunlai","middleName":"","lastName":"Zhu","suffix":""},{"id":291927747,"identity":"1e123ea8-905e-46a6-b7b8-3179c3fbc8b4","order_by":9,"name":"Zuheng Wu","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Zuheng","middleName":"","lastName":"Wu","suffix":""},{"id":291927753,"identity":"ad12071f-bd9e-44d3-a757-6666f337005b","order_by":10,"name":"Yuehua Dai","email":"","orcid":"","institution":"Anhui University","correspondingAuthor":false,"prefix":"","firstName":"Yuehua","middleName":"","lastName":"Dai","suffix":""}],"badges":[],"createdAt":"2024-04-07 07:30:30","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4230145/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4230145/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11128-025-04813-7","type":"published","date":"2025-06-23T15:57:13+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":55008706,"identity":"c630f3b0-2e79-4ede-b88e-02501170065f","added_by":"auto","created_at":"2024-04-19 19:07:39","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":209640,"visible":true,"origin":"","legend":"\u003cp\u003eComparison between serial QCNN (a) and parallel QCNN (b). (c) Illustration of the overall parallel QCNN framework, wherein the final fully connected layer of the base architecture is substituted with a parallel structure comprising classical and quantum components. The Input Layer functions as the point of entry for classical data, while the Classical Layers and Quantum Layers are employed for classical and quantum feature extraction, respectively. The ultimate fully connected layer is responsible for the output of multi-class classification tasks.\u003c/p\u003e","description":"","filename":"floatimage1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/650ba52caa0ad996d8b1c1c6.jpg"},{"id":55008709,"identity":"65e1b6e2-a762-48c9-9a64-4d74731b3cc4","added_by":"auto","created_at":"2024-04-19 19:07:39","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":115703,"visible":true,"origin":"","legend":"\u003cp\u003eThe quantum circuit used in our PQCNN with one layer\u003c/p\u003e","description":"","filename":"floatimage2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/13752e47cf65f7660be5eb11.jpg"},{"id":55009726,"identity":"f84db224-4fcc-45bd-97ab-d751f1d53c9d","added_by":"auto","created_at":"2024-04-19 19:15:39","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":729754,"visible":true,"origin":"","legend":"\u003cp\u003eCucumber plant disease dataset. (a) The left panel displays the original image of a healthy leaf, while the right panel exhibits the same image with added Gaussian noise of intensity 0.5. (b) The left panel illustrates the original image of a diseased leaf, with the right panel showcasing the image perturbed by Gaussian noise with intensity 1.0.\u003c/p\u003e","description":"","filename":"floatimage3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/19a262b2df6dfc0e56f05a3c.jpg"},{"id":55008707,"identity":"01224dc8-7e8b-4e63-aa89-3e025b9c48cc","added_by":"auto","created_at":"2024-04-19 19:07:39","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":185492,"visible":true,"origin":"","legend":"\u003cp\u003e(a) and (b) The accuracy and loss values comparison between the hybrid serial network and various configurations of hybrid parallel networks for binary classification using the cucumber dataset during training process. In particular, \"Parallel-QCQC\" denotes a configuration where the first and third layers are quantum, while the second and fourth layers are classical. (c) A graph comparing the accuracy of the hybrid serial structure (blue curve) with the hybrid parallel structure QCQC (red curve) across different noise intensities.\u003c/p\u003e","description":"","filename":"floatimage4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/5229b41380d715478b0fece7.jpg"},{"id":55009727,"identity":"797e2a0e-6f98-416a-8a51-72d41d3e8548","added_by":"auto","created_at":"2024-04-19 19:15:39","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":420497,"visible":true,"origin":"","legend":"\u003cp\u003eThe MNIST handwritten dataset used for five-class (a) and ten-class (b) classification tasks.\u003c/p\u003e","description":"","filename":"floatimage5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/d6bc9eaafc2983fa3c3e9663.jpg"},{"id":55009728,"identity":"c419758c-82fd-427f-906a-321e13f0c47f","added_by":"auto","created_at":"2024-04-19 19:15:39","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":329495,"visible":true,"origin":"","legend":"\u003cp\u003ePerformance comparison between hybrid serial networks and hybrid parallel networks with different configurations for both 5-class and 10-class classification tasks using the MNIST dataset.\u003c/p\u003e","description":"","filename":"floatimage6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/20a49411283e248dc6713812.jpg"},{"id":55008712,"identity":"b3c09755-5108-462e-bee3-716640412b07","added_by":"auto","created_at":"2024-04-19 19:07:39","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":308820,"visible":true,"origin":"","legend":"\u003cp\u003eThe confusion matrix of the two network structures for comparison. The confusion matrix above pertains to the MNIST five-class classification, while the one below corresponds to the MNIST ten-class classification.\u003c/p\u003e","description":"","filename":"floatimage7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/1495b7b7068c4372c1c7fcb6.jpg"},{"id":55008710,"identity":"f4e7b640-e7c7-43c5-9587-f65bfeba0ad5","added_by":"auto","created_at":"2024-04-19 19:07:39","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":94613,"visible":true,"origin":"","legend":"\u003cp\u003eThe graphs (a) and (b) depict the trends of accuracy variation with different numbers of quantum bits and different circuit depths, where the horizontal axes of graphs (a) and (b) represent different numbers of quantum bits and different circuit depths, respectively.\u003c/p\u003e","description":"","filename":"floatimage8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/4ceee07f60498abd0014f597.jpg"},{"id":85686114,"identity":"064a6ddf-b71e-4907-983f-67b2dd171a3c","added_by":"auto","created_at":"2025-06-30 16:03:34","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3260271,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4230145/v1/089e352b-7ddd-4e53-ba31-87ad9a9e272f.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Parallel Structure of Hybrid Quantum-Classical Neural Networks for Image Classification","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eThe convergence of quantum computing and artificial intelligence (AI) has sparked the emergence of a promising paradigm known as Quantum Neural Networks (QNNs)[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. QNNs represent a fusion of quantum computing principles with the computational framework of artificial neural networks, offering the potential to revolutionize both fields and tackle computationally intractable problems beyond classical techniques [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. The significance of QNNs lies in their capability to address complex optimization tasks across various domains such as optimization, cryptography, and machine learning. By harnessing the inherent parallelism and entanglement of quantum systems, QNNs offer the prospect of achieving exponential speedup in solving complex optimization tasks compared to classical algorithms [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Moreover, QNNs require fewer resources compared to classical AI models and benefit from the advantages of parallel quantum computing architectures [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHowever, implementing QNNs on current noisy intermediate-scale quantum (NISQ) devices poses significant challenges. Issues such as barren plateau problems and the curse of dimensionality hinder their practical realization [\u003cspan additionalcitationids=\"CR8 CR9\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. To address these challenges, hybrid quantum-classical architectures, termed Hybrid Quantum-Classical Neural Networks (HQCNNs), have emerged as a promising avenue for AI development. These models combine relatively small, realizable quantum circuits with classical neural networks [\u003cspan additionalcitationids=\"CR12\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Notably, various studies have extended classical machine learning algorithms into the quantum domain, with examples including quantum support vector machines [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], quantum nearest-neighbor algorithms [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], and quantum decision tree classifiers [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. Additionally, inspired by kernel methods in classical machine learning, quantum kernel methods have been proposed, utilizing quantum devices to estimate kernel matrices and offloading subsequent computations to classical computers [\u003cspan additionalcitationids=\"CR19\" citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eDespite these advancements, existing serial quantum-classical neural networks (SQCNNs), where information flows sequentially between quantum and classical networks, face limitations that affect network expressivity and trainability [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. For instance, SQCNNs are susceptible to input perturbations and exhibit decreased accuracy for multi-class image classification tasks, especially in the presence of noisy input data [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. In this study, we address these challenges by introducing a novel Hybrid Parallel Quantum Convolutional Neural Network (PQCNN) for image classification. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(a) and Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b) illustrate a comparison between the traditional quantum serial network and our proposed quantum parallel network. By parallelizing quantum layers with classical layers, our aim is to integrate the advantages of quantum and classical computing, thereby constructing a more efficient and powerful network model. Our contributions in this work are as follows:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eWe introduce the concept of Hybrid Parallel Quantum Convolutional Neural Network (PQCNN), which combines quantum and classical computing in a parallel manner, facilitating its application in tasks such as image classification.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eWe detail the structure of the PQCNN model, elucidating the design of quantum and classical layers and their interconnections. Additionally, we explore the impact of multi-layer parallel structures on experimental results.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eWe demonstrate the effectiveness of PQCNN using various datasets and classification scenarios, investigating its robustness by introducing noise. We analyze the effects of different numbers of quantum bits and circuit depths on the experiments. Moreover, we conduct comparative experiments to validate the efficient performance of our model.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eThe remainder of the paper is organized as follows: Chap.\u0026nbsp;2 delves into Related Work, discussing previous research in hybrid classical-quantum machine learning and quantum transfer learning. Chapter\u0026nbsp;3 presents our Proposed Model and Quantum Classifier, providing a comprehensive overview of the model's architecture and design. Chapter\u0026nbsp;4 showcases our Experimental Results, conducting simulations using the Cucumber dataset and the MNIST dataset to validate the effectiveness of our proposed PQCNN model. Finally, in Chap.\u0026nbsp;5, we draw conclusions, summarize key findings, and reflect on implications and potential future directions in the field of hybrid quantum-classical machine learning.\u003c/p\u003e"},{"header":"2 Related works","content":"\u003cp\u003eNumerous significant researches have delved into the fusion of quantum computing and machine learning, with the objective of leveraging quantum algorithms to enrich conventional learning approaches. This section provides a succinct overview of the pertinent studies on the quantum computing framework in image classification, along with the ongoing research trajectory and challenges encountered.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Hybrid Classical-Quantum Machine Learning\u003c/h2\u003e \u003cp\u003eHybrid classical-quantum machine learning represents a burgeoning field at the intersection of classical and quantum computing, aiming to combine the strengths of both paradigms to tackle complex computational tasks more efficiently. In this approach, classical machine learning techniques are integrated with quantum algorithms to leverage the computational advantages offered by quantum computing while mitigating the limitations of current quantum hardware. Notable research efforts in this area include the Hybrid Quantum-Classical Neural Network (QCNN) proposed by Liu et al., which combines classical neural networks with variational quantum circuits to enhance learning capabilities [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. Similarly, the Quantum Variational Classifier (QVC) introduced by Adhikary et al. (2020) utilizes classical optimization algorithms to train variational quantum circuits for classification tasks, showcasing the potential synergy between classical and quantum methodologies [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. Moreover, hybrid approaches such as Quantum Enhanced Support Vector Machines (QSVM) and Quantum Generative Adversarial Networks (QGAN) have demonstrated promising results in various machine learning tasks [\u003cspan additionalcitationids=\"CR27\" citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eDespite the challenges posed by the current limitations of quantum hardware, such as noise and decoherence, hybrid classical-quantum machine learning holds great promise for unlocking new capabilities and achieving breakthroughs in solving complex problems across diverse domains. Ongoing research efforts focus on refining hybrid algorithms, developing novel techniques for integrating classical and quantum components, and exploring applications in areas such as optimization, pattern recognition, and data analysis. As quantum technology continues to advance, hybrid classical-quantum machine learning is poised to play a pivotal role in shaping the future of computational science and artificial intelligence.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 QCNN for image classification with serial structure\u003c/h2\u003e \u003cp\u003eQuantum Classical Neural Networks (QCNNs) signify a compelling amalgamation of classical and quantum computing principles tailored for image classification endeavors. By harnessing the complementary attributes of classical and quantum elements, these hybrid architectures endeavor to exploit the inherent advantages of each paradigm. Typically, this entails the orchestration of quantum circuits in tandem with classical multi-layer perceptron (MLP) structures. For instance, Mari et al. [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e] elucidated diverse approaches to hybrid transfer learning, amalgamating classical transfer learning techniques with quantum neural networks for image classification tasks. This strategy not only leverages the computational prowess of quantum computers but also integrates methodologies proven successful in classical machine learning, thereby showcasing substantial potential and utility. Furthermore, Ref. [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e] proposed a scale-inspired local feature extraction technique grounded in Quantum Convolutional Neural Networks, demonstrating enhancements in both recognition and classification accuracy. Additionally, Konar et al. introduced a shallow hybrid QCNNs framework aimed at addressing robust image classification challenges amidst noise and adversarial attacks [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. However, it's noteworthy that the majority of QCNN structures are inherently serial, a characteristic that significantly amplifies trainability and expressivity [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e], albeit at the cost of heightened susceptibility to overfitting and increased computational complexity.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 PQCNN\u003c/h2\u003e \u003cp\u003eIn classical neural networks, parallel structures have been extensively employed in complex architectures such as Inception [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e], ResNet [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e], and DenseNet [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e]. Unlike the serial approach, parallel structures offer enhanced computational efficiency and expedited training times, effectively capturing features across multiple scales and abstraction levels. These advantages render the parallel structure of neural networks promising for the advancement of machine learning, a potential extension to QCNN.\u003c/p\u003e \u003cp\u003eSeveral researches have shown that the serial architecture of Quantum Convolutional Neural Networks (QCNN) introduces an information bottleneck, directly impeding the network's capacity for representation and impacting its expressive power and performance. As referenced in [\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e], Kordzanganeh et al. proposed the Parallel QCNN, which integrates Multi-Layer Perceptron (MLP) and Variational Quantum Circuits (VQCs) in parallel. It has been found that PQCNN can effectively extract both harmonic and non-harmonic features from datasets. Leveraging its unique architecture, PQCNN demonstrates the ability to discern complex patterns and relationships within data, a task that traditional machine learning algorithms may find challenging. This work underscores the potential of parallel structures in QCNN to enhance performance across a broad spectrum of tasks. Currently, there is limited research on parallel network layers in QCNN, and its applicability to image classification tasks remains unexplored, warranting further investigation.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"3 Method","content":"\u003cp\u003eThis paper aims to develop a hybrid quantum-classical classifier tailored for multi-class classification tasks, utilizing a parallel network structure. Specifically, we employ a classical-to-quantum transfer learning approach for image classification, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Our selected classical network is ResNet18, a renowned Convolutional Neural Network (CNN) model esteemed for its efficacy in classification tasks and pre-trained on the dataset.\u003c/p\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Architecture of PQCNN\u003c/h2\u003e \u003cp\u003eOur research paper is based on a transfer learning architecture, serving as the foundational framework for our work. Within this framework, we adopt a hybrid parallel classical-quantum circuit, where the final fully connected layer of the base architecture is replaced with a parallel structure for multi-class classification. This parallel combination of classical and quantum components forms the core of our data processing and classification approach, allowing for multi-layer parallelization. The specific framework diagram is depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Quantum Layer\u003c/h2\u003e \u003cp\u003eIn this section, we provide a detailed explanation of the data encoding method, quantum circuit structure, and data decoding process of the quantum-classical multi-class classifier.\u003c/p\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e3.2.1 Data Encoding\u003c/h2\u003e \u003cp\u003eData encoding refers to the process of converting data from one format or representation into another format suitable for transmission, storage, or processing. In the context of quantum computing or quantum information processing systems, data encoding involves converting classical information into quantum states that can be manipulated and processed using quantum algorithms.\u003c/p\u003e \u003cp\u003eIn this paper, we adopt an angle-based encoding scheme for the quantum-classical parameter mapping, converting classical parameters into quantum parameters and then transforming them back into classical parameters as output [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e]. For a classical input data \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x{=[{x}_{1},\\dots ,{x}_{n}]\\mathbb{ }\\in \\mathbb{R}}^{n}\\)\u003c/span\u003e\u003c/span\u003e, We utilize angle encoding to transform the input \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x\\)\u003c/span\u003e\u003c/span\u003e to a quantum state \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left|x\\right.⟩\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left|x\\right.⟩\\)\u003c/span\u003e\u003c/span\u003e can be written as:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\left|x\\right.⟩= {\\otimes }_{i=1}^{N}\\text{cos}\\left({x}_{i}\\right)\\left|0\\right.⟩+\\text{sin}\\left({x}_{i}\\right)\\left|1\\right.⟩$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAngle encoding utilizes \u003cem\u003eN\u003c/em\u003e qubits along with a constant-depth quantum circuit for the encoding process. The state preparation unitary, denoted as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S}_{{x}_{j}}\\)\u003c/span\u003e\u003c/span\u003e, is expressed as the tensor product of individual unitary operators \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({U}_{i}\\)\u003c/span\u003e\u003c/span\u003e, where \u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, 2, \u0026hellip;, N. Each \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({U}_{i}\\)\u003c/span\u003e\u003c/span\u003e can be written as follows:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${U}_{i}=\\left[\\begin{array}{cc}\\text{c}\\text{o}\\text{s}\\left({x}_{j}^{\\left(i\\right)}\\right)\u0026amp; -\\text{s}\\text{i}\\text{n}\\left({x}_{j}^{\\left(i\\right)}\\right)\\\\ \\text{s}\\text{i}\\text{n}\\left({x}_{j}^{\\left(i\\right)}\\right)\u0026amp; -\\text{c}\\text{o}\\text{s}\\left({x}_{j}^{\\left(i\\right)}\\right)\\end{array}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}^{\\left(i\\right)}\\)\u003c/span\u003e\u003c/span\u003e represents the angle associated with the \u003cem\u003ej\u003c/em\u003e-th parameter of the classical data being encoded, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({U}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the corresponding unitary operator acting on the \u003cem\u003ei\u003c/em\u003e-th qubit. This formulation enables the encoding of classical parameters into quantum states using angles, facilitating efficient processing within quantum circuits or algorithms.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e \u003ch2\u003e3.2.2 Variational Quantum Circuits\u003c/h2\u003e \u003cp\u003eA quantum circuit comprises a sophisticated arrangement of quantum gates, fundamental units for manipulating qubits, akin to classical logic gates. By organizing quantum gates in a specific sequence, intricate circuits are formed to execute diverse quantum algorithms and tasks. Notable quantum gates, including the Hadamard gate, CNOT gate, and phase gate, facilitate qubit interactions, quantum superposition state generation, and manipulation, among other functionalities [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e]. Quantum gates within these circuits are typically unitary, ensuring reversibility for preserving and transmitting quantum information.\u003c/p\u003e \u003cp\u003eOur circuit features single-qubit \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{R}}_{\\text{y}}\\left({\\theta }\\right)\\)\u003c/span\u003e\u003c/span\u003e g gates and two-qubit CNOT gates, constituting a hardware-efficient setup for network training, as depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The formulas representing the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{R}}_{\\text{y}}\\left({\\theta }\\right)\\)\u003c/span\u003e\u003c/span\u003e gate and the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{C}\\text{N}\\text{O}\\text{T}\\)\u003c/span\u003e\u003c/span\u003e gate matrix representation are given as:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\text{C}\\text{N}\\text{O}\\text{T}=\\left[\\begin{array}{cc}\\begin{array}{cc}1\u0026amp; 0\\\\ 0\u0026amp; 1\\end{array}\u0026amp; \\begin{array}{cc}0\u0026amp; 0\\\\ 0\u0026amp; 0\\end{array}\\\\ \\begin{array}{cc}0\u0026amp; 0\\\\ 0\u0026amp; 0\\end{array}\u0026amp; \\begin{array}{cc}1\u0026amp; 0\\\\ 0\u0026amp; 1\\end{array}\\end{array}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${\\text{R}}_{\\text{y}}\\left({\\theta }\\right)=\\left[\\begin{array}{cc}\\text{c}\\text{o}\\text{s}\\left(\\frac{{\\theta }}{2}\\right)\u0026amp; -\\text{s}\\text{i}\\text{n}\\left(\\frac{{\\theta }}{2}\\right)\\\\ \\text{s}\\text{i}\\text{n}\\left(\\frac{{\\theta }}{2}\\right)\u0026amp; \\text{c}\\text{o}\\text{s}\\left(\\frac{{\\theta }}{2}\\right)\\end{array}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe Controlled-NOT (CNOT) gate entangles qubits and incorporates a control parameter, θ, which assists in the training process and accelerates model speed by enabling parallel computation. A variational quantum circuit of depth q is a succession of numerous quantum layers, each representing the product of multiple units parameterized by distinct weight values. This concept can be mathematically expressed as:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${\\mathcal{M}={L}_{1}\\cup {L}_{2}\\cup \\bullet \\bullet \\bullet \\cup L}_{\\text{q}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mathcal{M}\\)\u003c/span\u003e\u003c/span\u003e represents the overall structure of the quantum circuit, where \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e denotes one layer of the quantum circuit as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The notation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{L}_{1}\\cup {L}_{2}\\cup \\bullet \\bullet \\bullet \\cup L}_{\\text{q}}\\)\u003c/span\u003e\u003c/span\u003e represents the concatenation of q layers of the circuit.\u003c/p\u003e \u003cp\u003eThe depth of the quantum circuit is also a crucial parameter, representing the number of layers of quantum gates cascaded together. Deeper quantum circuits typically enable the execution of more complex computational tasks, but they also require more qubits and gate operations, necessitating a balance between depth and resource consumption when designing quantum circuits.\u003c/p\u003e \u003cp\u003eQuantum circuits serve as the core tools in quantum computing. By strategically designing and optimizing the connections and parameters of quantum gates, various complex quantum computing tasks can be accomplished.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section3\"\u003e \u003ch2\u003e3.2.3 Data Decoding\u003c/h2\u003e \u003cp\u003eThe decoding layer plays a pivotal role in converting the quantum state into classical states, commonly accomplished through measurement operations. During this process, measurement is conducted utilizing the Pauli-Z operator to acquire the expected value of the Z-direction spin component of the qubit, thereby producing the corresponding classical state vector. The specific expression for this operation is as follows:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\mathcal{F}=\\left|X\\right.⟩\\to Y=⟨X\\left|\\stackrel{-}{Y}\\right|X⟩$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left|X\\right.⟩\\)\u003c/span\u003e\u003c/span\u003e represents the input quantum state, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Y\\)\u003c/span\u003e\u003c/span\u003e represents the decoded classical state, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\stackrel{-}{Y}\\)\u003c/span\u003e\u003c/span\u003e denotes the measurement operator used for decoding.\u003c/p\u003e \u003cp\u003eThe objective of this process is to extract classical information from the quantum state, typically represented by the expectation value associated with a specific physical quantity. Hence, the decoding layer assumes a critical role in quantum computing, facilitating the conversion of quantum information into classical information for subsequent processing and analysis.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Classical Layer\u003c/h2\u003e \u003cp\u003eIn the classical layer segment of the parallel architecture, we implemented two fully connected layers for transition purposes. Each neuron within the fully connected layer establishes complete connectivity with all neurons in its preceding layer. This design allows the fully connected layer to amalgamate localized information containing category-specific discriminative features extracted from convolutional or pooling layers. Furthermore, at the network's culmination, we employed a fully connected layer to serve as the output mechanism for classification, essentially functioning as a sophisticated \"classifier\" within our comprehensive network framework.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Loss Function and Evaluation of the PQCNN\u003c/h2\u003e \u003cp\u003eIn the training process of our PQCNN, we employ the cross-entropy loss function to facilitate the classification task. This pivotal metric allows us to measure the dissimilarity between the predicted outcomes generated by our model and the actual ground truth labels associated with the data. The cross-entropy loss function is expressed mathematically as follows:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$Loss=\\frac{1}{N}\\sum _{i}(-\\sum _{j=1}^{M}{y}_{ij}\\text{l}\\text{o}\\text{g}\\left({p}_{ij}\\right))$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eM\u003c/em\u003e represents the number of categories, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}_{ij}\\)\u003c/span\u003e\u003c/span\u003e represents whether the sample \u003cem\u003ei\u003c/em\u003e belongs to category \u003cem\u003ej\u003c/em\u003e (1 if it does, 0 otherwise), and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{ij}\\)\u003c/span\u003e\u003c/span\u003e represents the predicted probability that observation \u003cem\u003ei\u003c/em\u003e pertains to category \u003cem\u003ej\u003c/em\u003e.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Experimental Results and Analysis","content":"\u003cp\u003eIn this section, we delve into the intricacies of our experiments and simulations, showcasing the efficacy of our method across two diverse datasets: the cucumber plant disease dataset and the handwritten digit dataset (MNIST), both utilized for classification tasks. We meticulously examine the results obtained, providing comprehensive insights into the performance and robustness of our approach.\u003c/p\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Experimental Environment and Training Setup\u003c/h2\u003e \u003cp\u003eOur experimental setup was executed on a local computing system featuring a ten-core CPU clocked at 2.5 GHz. The computational framework employed for our experiments comprised PennyLane and PyTorch. PennyLane, an open-source Python library, enables seamless integration of quantum and classical computing through automatic differentiation. This framework seamlessly interfaces with popular machine learning libraries like TensorFlow and PyTorch.\u003c/p\u003e \u003cp\u003eDuring the training phase, we employed the Adam optimizer with a learning rate of 0.0004 for 15 iterations. The quantum component of our model was composed of 4 qubits, while the circuit depth was set to 6 layers.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Evaluation of Our Methodology Using Cucumber Plant Disease Dataset\u003c/h2\u003e \u003cp\u003eOur research focuses on evaluating the effectiveness of our methodology for binary classification employing the cucumber plant disease dataset. This dataset constitutes a vital segment within a comprehensive repository of plant disease data extensively employed in scientific investigations. It encompasses a diverse array of images showcasing different diseases afflicting cucumber plants. The principal purpose of curating this dataset is to expedite the advancement and optimization of machine learning algorithms tailored for automated detection and characterization of plant diseases.\u003c/p\u003e \u003cp\u003eTo thoroughly examine the performance of our proposed methodology, we conducted experiments involving the introduction of Gaussian noise with different level. Gaussian noise, known for its normal distribution, effectively replicates random disturbances encountered in real-world scenarios [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e]. During the experimentation phase, we manipulated the variance or standard deviation of the Gaussian noise to modulate its intensity. By injecting Gaussian noise at varying levels into the cucumber plant disease dataset, we were able to assess the model's efficacy amidst differing degrees of interference, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. This enabled a comprehensive evaluation of its performance under diverse environmental conditions.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(a) and Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(b) illustrate the accuracy and loss metrics extracted from binary classification tasks conducted on the cucumber plant disease dataset, employing a range of network architectures. These visual representations offer valuable insights into the performance dynamics observed across diverse models within the dataset. We conducted experiments utilizing neural networks featuring distinct parallel structures, including parallel quantum-classical (QC), quantum-classical-quantum (QCQ) configurations, among others. Our findings underscore a notable superiority in accuracy for the parallel structure QCN compared to its traditional serial counterpart. For instance, the parallel-QCN achieves an accuracy of 96.1%, contrasting with approximately 94.5% accuracy for the serial QCN. The loss function curves presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(b) further elucidate these observations, revealing that the parallel structure's convergence values are consistently smaller than those of the serial structure. These outcomes strongly indicate that parallel structures exhibit superior image recognition capabilities relative to their serial counterparts in the context of QCNs.\u003c/p\u003e \u003cp\u003eAfter introducing Gaussian noise into the data images, the superiority of the hybrid parallel quantum convolutional neural network over its traditional serial counterpart becomes apparent. Despite a marginal decrease in accuracy with escalating noise intensity, the parallel architecture consistently maintains a commendable level of precision. In contrast, the accuracy of the serial configuration exhibits a sharp downward trajectory, as depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(c). This outcome underscores the superior robustness of the hybrid parallel architecture in mitigating noise interference. The parallel structure's efficacy arises from its ability to concurrently process information from multiple branches, thereby offering heightened resistance against the adverse impact of noise on model performance. Conversely, the serial structure is more susceptible to interference in the presence of noise, resulting in a precipitous decline in accuracy. The high accuracy and resilience of PQCNN presumably stem from its streamlined training process leveraging parallel computing and the formidable anti-jamming capability inherent in quantum neural networks as indicated in [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eMoreover, the parallelization of multiple quantum or classical layers presents an opportunity to augment the model's capacity and expressive prowess. Envisioning the design of such parallelized structures with multiple quantum or classical layers holds promise for further enhancing the model's robustness and performance, rendering it more dependable and efficient in managing noise and intricate data. Consequently, harnessing the advantages of parallel structures comprehensively during the model's design and optimization stages can furnish us with increased options and adaptability, thereby culminating in superior model performance.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e4.3 MNIST Handwritten Digit Dataset for Multi-Class Image Classification\u003c/h2\u003e \u003cp\u003eIn this investigation, we conducted an extensive evaluation of the effectiveness of our proposed methodology within the domain of handwritten digit recognition, using the widely recognized MNIST dataset as our experimental platform. Renowned for its consistency and applicability, the MNIST dataset consists of 70,000 grayscale images, each meticulously labeled to represent handwritten digits spanning from 0 to 9. To streamline our analysis, we meticulously curated a subset from this extensive dataset, ensuring an equitable distribution of 1000 images for each digit category. Subsequently, this subset was divided into distinct training and testing sets, with 800 images allocated for training and the remaining 200 images reserved for rigorous testing.\u003c/p\u003e \u003cp\u003eIn framing our experimental approach, we delineated two distinct classification tasks: a five-class classification task and a ten-class classification task. For the former, our focus was specifically on digits 3 through 7, resulting in a dataset comprising five thousand images representing this narrower range of handwritten numerals. In contrast, the latter classification task encompassed the entirety of the MNIST dataset, spanning all ten digits (0 through 9) and yielding a comprehensive collection of 10,000 images. To offer visual insights into our experimental configuration, we present representative samples from the 5-classification and 10-classification tasks in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(a) and Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(b) respectively. These visualizations provide a glimpse into the diversity and intricacy inherent in the MNIST dataset, reaffirming its pivotal role as a benchmark for assessing the performance of machine learning algorithms in the realm of handwritten digit recognition.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn the initial phase of our research, we conducted a thorough examination of the MNIST dataset by employing various parallel multi-layer configurations. We addressed both five-classification and ten-classification tasks to explore the dataset comprehensively. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates the performance trajectories, detailing accuracy and loss for each task category. Our analysis revealed a significant performance gap between the QCQCQ model implemented in the parallel multi-layer architecture and its serial counterpart.\u003c/p\u003e \u003cp\u003eFor the five-classification task, the QCQCQ model exhibited an impressive accuracy of 95.90%, surpassing the accuracy of the serial structure by 17.30%, which stood at 78.6%. This notable improvement underscores the efficacy of the parallel multi-layer architecture. Similarly, in the context of the ten-classification task, the QCQCQ model achieved an accuracy of 93.90%, showcasing a substantial enhancement of 42.62% compared to the serial structure's accuracy of 51.25%. These findings highlight the superiority of the parallel multi-layer configuration in handling complex classification tasks.\u003c/p\u003e \u003cp\u003eThe elucidation provided by Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, showcasing the intricate details of the confusion matrix, offers a profound understanding of the predictive capabilities across individual digits. It meticulously dissects the performance disparities among the hybrid serial CQ, hybrid parallel QC, and hybrid parallel QCQC architectures. Upon scrutinizing the five-class MNIST image classification confusion matrix, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(a), glaring differences emerge. The serial-CQ structure struggles to correctly identify a mere 22 instances out of 200 for the digit \u0026ldquo;3\u0026rdquo;, starkly contrasting with the commendable performance of the hybrid parallel QC and QCQC configurations, which accurately classify 179 and 183 instances of the digit \"3\", respectively.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe summary of data for binary, five-class, and ten-class classification tasks\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDataset\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStructure\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAccuracy\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePrecision\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eRecall\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eF1-Score\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"4\" rowspan=\"5\"\u003e \u003cp\u003eCucmber-2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSerial-CQ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9459\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9446\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9462\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9454\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9639\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9628\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9645\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9636\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-CQC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9684\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9677\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9685\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9681\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QCQC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e0.9774\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003e0.9768\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e0.9777\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e0.9772\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QCQCQ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9639\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9623\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9664\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9637\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"4\" rowspan=\"5\"\u003e \u003cp\u003eMnist-5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSerial-CQ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.786\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.7942\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.786\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.7413\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.937\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9369\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9369\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9366\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-CQC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.955\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9554\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.955\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9548\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QCQC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.955\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9550\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9549\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9549\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QCQCQ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e0.959\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003e0.9590\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e0.959\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e0.9588\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"4\" rowspan=\"5\"\u003e \u003cp\u003eMnist-10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSerial-CQ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5125\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.4936\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.5125\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.4371\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.932\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9322\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.932\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9319\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-CQC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9365\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9364\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9365\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9362\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QCQC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9375\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9366\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.9359\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.9359\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParallel-QCQCQ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e0.939\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003e0.9398\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e0.939\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e0.939\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eA comprehensive summary of the experimental results is encapsulated in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. In addition to the accuracy advantages for the parallel-QCN compared with the serial structure, the parallel-QCN also outperforms the serial model across all metrics, including the precision, recall, and F1-score. The superior performance of the parallel-QCN model can be attributed to its ability to capture more information and features present in the images through the quantum and classical layers, as discussed in Ref.[\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e]. In contrast, the serial-QCN model, although capable of capturing sequential information, may cause the information bottlenecks, limiting the expressivity of the network.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn our exploration of quantum circuit dynamics in relation to classification tasks, we delved into the interplay between the number of qubits and circuit depth, employing the MNIST dataset for a ten-classification task. Initially, with the circuit depth held constant at 6, we conducted experiments varying the number of qubits from 2 to 10. The resultant insights, depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e(a), highlighted a notable trend: as the number of qubits increased, experimental accuracy exhibited a progressive enhancement. This phenomenon underscores the pivotal role of qubit quantity in augmenting the quantum circuit's representational prowess. Indeed, by affording greater degrees of freedom, an increased qubit count enables the circuit to adeptly capture the intricate features inherent within the dataset. Conversely, a dearth of qubits may curtail the circuit's capacity to effectively encode complex data patterns. Thus, the observed improvement in accuracy with escalating qubit numbers is emblematic of the circuit's augmented expressive power, facilitating a finer delineation of the dataset's nuanced structure.\u003c/p\u003e \u003cp\u003eSubsequently, with the number of qubits set at 4, we scrutinized the impact of varying circuit depth from 2 to 10. The resultant observations, delineated in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e(b), revealed an intriguing relationship between depth and classification accuracy. Remarkably, the pinnacle of accuracy was attained at a circuit depth of 4. This finding underscores the delicate balance required in determining an optimal circuit depth. Insufficient depth risks the specter of underfitting, where the circuit fails to capture the intricacies of the dataset, thereby compromising classification accuracy. Conversely, an excessive depth engenders the pernicious encroachment of noise within the circuit []. As the depth amplifies, the concomitant proliferation of qubit interactions may undermine coherence between quantum states, culminating in heightened noise levels detrimental to computational fidelity. Thus, in navigating the labyrinthine terrain of circuit depth optimization, considerations spanning dataset complexity, noise mitigation strategies, and computational resource constraints emerge as pivotal determinants in fostering the zenith of classification performance.\u003c/p\u003e \u003cp\u003eIn synthesis, our investigation into the nexus of qubit quantity and circuit depth illuminates the nuanced dynamics shaping the efficacy of quantum circuits in classification tasks. By discerning the intricate interplay between these parameters, we endeavor to chart a course towards unlocking the full potential of quantum computational paradigms in the realm of machine learning and beyond.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Comparative Analysis\u003c/h2\u003e \u003cp\u003eThe comparative analysis conducted in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e highlights the significant advantages offered by our method over alternative quantum approaches in the realm of 5-class and 10-class MNIST image classification. Notably, our method outperforms quantum neural networks referenced in Ref. [\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e] and [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e], as well as serial quantum-classical neural networks mentioned in Ref. [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e]. However, it's crucial to acknowledge that the data presented in the table are sourced from various studies, each utilizing distinct dataset specifications. Direct comparisons between methodologies must be approached with caution due to potential discrepancies in dataset composition and characteristics. These variations can obscure the interpretation of results and may lead to misleading conclusions. Therefore, it's essential to view these findings as contextual reference points rather than definitive assessments.\u003c/p\u003e \u003cp\u003eWhile our method demonstrates clear advantages, it's imperative to recognize the complexity of the broader quantum landscape. A nuanced scrutiny of different quantum methodologies is necessary, taking into account the intricacies inherent in dataset variability and experimental design. By doing so, we can gain valuable insights into the relative efficacy of various quantum approaches within their respective experimental contexts, thus advancing the field of quantum image classification.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComparison of accuracy with other quantum methods\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDataset\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAccuracy\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOur method\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMnist(3\u0026ndash;7)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e95.9%\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMnist (0\u0026ndash;9)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e93.9%\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQNN Ref.[\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMnist (0\u0026ndash;9)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e65.13%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQNN Ref.[\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMnist (3\u0026ndash;6)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e85.14%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQCNN Ref.[\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMnist (0\u0026ndash;9)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e79.9%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5 Summary","content":"\u003cp\u003eIn conclusion, our paper introduces the Hybrid Parallel Quantum Classical Neural Network (PQCNN), a novel approach aimed at overcoming challenges encountered in implementing serial Quantum Classical Neural Networks (SQCNN) for image classification tasks. Through a thorough exploration of the PQCNN model, backed by detailed experimental results and analysis, several significant findings have been elucidated. The experimental validation conducted across datasets, including the cucumber plant disease dataset and the MNIST handwritten digit dataset, highlights the effectiveness of PQCNN in achieving high accuracy and robust performance in classification tasks. Analysis of various performance metrics, including accuracy, precision, recall, and F1-score, emphasizes the superior capabilities of PQCNN over SQCNN, particularly in addressing complex classification tasks across diverse datasets.\u003c/p\u003e \u003cp\u003eThese findings provide valuable insights into optimizing quantum-classical neural network architecture, with considerations ranging from noise mitigation to multiclassification scenarios. Overall, our results contribute to advancing the understanding and application of quantum computing in image classification tasks through the development and validation of the PQCNN model. This research opens avenues for further exploration and investigation in the realm of hybrid quantum-classical machine learning, with potential implications extending beyond image classification to a wide array of applications.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eZ. Xu and Y. Hu wrote the main manuscript and conceptualized the study, T. Yang, P. cai, K. Shen, and B. Lv participated in data collection and analysis, S. Chen, J, Wang prepared figure 1-4 and checked the language, Y. Zhu and Z. Wu contributed to the writing and editing of the manuscript, Y. Dai supervised this work. All authors reviewed the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eThis work was partly supported by the National Natural Science Foundation of China (Grant Nos. 61874001, 62004001, 62201005, 62004001, 62304001), the Anhui Provincial Natural Science Foundation under Grant No. 2308085QF213, 2308085QF195, and the Natural Science Research Project of Anhui Educational Committee under Grant No. 2022AH050106, 2023AH050072.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe data that support the findings of this study are available from the corresponding author upon reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eSchuld, M., Sinayskiy, I., Petruccione, F.: The quest for a Quantum Neural Network. 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Phys. 10, 1069985 (2022). https://doi.org/10.3389/fphy.2022.1069985\u003c/li\u003e\n\u003cli\u003eKashyap, S., Garani, S.S.: Quantum Convolutional Neural Network Architecture for Multi-Class Classification. In: 2023 International Joint Conference on Neural Networks (IJCNN). pp. 1\u0026ndash;8. IEEE, Gold Coast, Australia (2023)\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"quantum-information-processing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qinp","sideBox":"Learn more about [Quantum Information Processing](http://link.springer.com/journal/11128)","snPcode":"11128","submissionUrl":"https://submission.nature.com/new-submission/11128/3","title":"Quantum Information Processing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Quantum computing, Quantum-Classical networks, Parallel Structure, Image classification","lastPublishedDoi":"10.21203/rs.3.rs-4230145/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4230145/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eHybrid quantum-classical neural networks (QCNNs) integrate principles from quantum computing principle and classical neural networks, offering a novel computational approach for image classification tasks. However, current QCNNs with sequential structures encounter limitations in accuracy and robustness, especially when dealing with tasks involving numerous classes. In this study, we propose a novel solution - the hybrid Parallel Quantum Classical Neural Network (PQCNN) - for image classification tasks. This architecture seamlessly integrates the parallel processing capabilities of quantum computing with the hierarchical feature extraction abilities of classical neural networks, aiming to overcome the constraints of conventional sequential structures in multi-class classification tasks. Extensive experimentation demonstrates the superiority of PQCNN over traditional concatenative structures in binary classification datasets, displaying heightened accuracy and robustness against noise. Particularly noteworthy is PQCNN's significantly improved accuracy on datasets with 5 and 10 classes. These findings underscore the transformative potential of the PQCNN architecture as an advanced solution for enhancing the performance of quantum-classical-based classifiers, particularly in the domain of image classification.\u003c/p\u003e","manuscriptTitle":"Parallel Structure of Hybrid Quantum-Classical Neural Networks for Image Classification","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-04-19 19:07:34","doi":"10.21203/rs.3.rs-4230145/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-10-08T11:20:50+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-06-07T11:59:06+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"3723397426403868435088196357976754717","date":"2024-06-07T11:43:46+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"229791141284512546811113916859140191188","date":"2024-06-03T11:43:51+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"316557799896515313820334505730931857666","date":"2024-05-29T07:36:05+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-05-28T11:25:33+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-04-08T14:03:00+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-04-08T14:02:59+00:00","index":"","fulltext":""},{"type":"submitted","content":"Quantum Information Processing","date":"2024-04-07T07:29:15+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"quantum-information-processing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qinp","sideBox":"Learn more about [Quantum Information Processing](http://link.springer.com/journal/11128)","snPcode":"11128","submissionUrl":"https://submission.nature.com/new-submission/11128/3","title":"Quantum Information Processing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"179174f9-f9ed-4b6b-819d-51a3c6a05c58","owner":[],"postedDate":"April 19th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-06-30T15:59:25+00:00","versionOfRecord":{"articleIdentity":"rs-4230145","link":"https://doi.org/10.1007/s11128-025-04813-7","journal":{"identity":"quantum-information-processing","isVorOnly":false,"title":"Quantum Information Processing"},"publishedOn":"2025-06-23 15:57:13","publishedOnDateReadable":"June 23rd, 2025"},"versionCreatedAt":"2024-04-19 19:07:34","video":"","vorDoi":"10.1007/s11128-025-04813-7","vorDoiUrl":"https://doi.org/10.1007/s11128-025-04813-7","workflowStages":[]},"version":"v1","identity":"rs-4230145","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4230145","identity":"rs-4230145","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}