Multiclass Classification using VariationalQuantum Circuit on Benchmark Dataset

Multiclass Classification using VariationalQuantum Circuit on Benchmark Dataset | 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 Article Multiclass Classification using VariationalQuantum Circuit on Benchmark Dataset Muhammad Hamid, Bashir Alam, Om Pal Pal This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6069218/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Today, classification is a significant task in data science, and many industries, including healthcare, transport, and banking sectors, are required to classify the data. In this NISQ era, quantum computers are capable of solving complex data challenges and can predict results with minimum features. The quantum neural network is being studied extensively for machine learning problems. In this paper, we have performed the multiclass classification using variational quantum circuits on benchmark datasets. A combination of quantum and classical neural networks is used to build the quantum circuit and optimize the parameters. The quantum circuit is used for the feedforward architecture, while in back-propagation, parameters are updated using a classical optimizer on classical computers. We have successfully shown classification using the proposed approach in benchmark datasets, such as the Iris flower and the MNIST Digit data set. Our results show that VQC is a promising candidate for classification problems with fewer features. To perform our experiments we have used IBM Quantum hardware and simulators. Physical sciences/Engineering Physical sciences/Mathematics and computing Neural Network Multi-class Classification Variational Quantum Circuits Hybrid Quantum Classical Algorithm Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1. Introduction We have seen massive growth in artificial intelligence in the past three decades. There are a wide range of applications of artificial intelligence in healthcare, agriculture, transportation, finance, the advertising industry, and many more. Generally, machine learning model training requires vast computational resources and time; sometimes training a deep learning model may take several days and even a month. Despite recent advances in classical computers, modern computers are reaching their limits in implementing machine learning in different areas. Researchers and computer scientists are looking for an alternative to classical computing, and they see quantum computers as an alternate solution to overcome this problem. The quantum computers have shown speedup in factorization of numbers [ 1 ] and searching in unstructured data [ 2 ]. Quantum computers use quantum phenomena called superposition, decoherence, and entanglement to process quantum data. QML is an emerging discipline, an intersection of quantum computing and artificial intelligence. Though QML is still in its initial stages, a variety of QML algorithms have still been proposed. QML has been studied for supervised, unsupervised, semi supervised, and reinforcement learning [ 3 – 5 ]. Today we have QSVM [ 6 ], quantum k nearest neighbor [ 7 ] for the binary classification task. Various studies have been done regarding supervised, semi-supervised, and unsupervised learning using quantum-classical-based machine learning algorithms [ 9 ]. D.K. Park et al. have proposed a kernel-based quantum classifier [ 8 ]. We also have witnessed the power of Hybrid-QCNN for multi-class classification tasks [ 10 ]. Adhikary et. al. [ 11 ] have used N level quantum systems to encode features for a Hybrid Quantum Classical classifier (HQC) for classification on different datasets. Davis Arthur and Prasanna Date[ 12 ] have proposed HQC neural network architecture for binary classification and [ 13 ] have demonstrated 3 class classification with high accuracy using amplitude encoding. Here we present a mixed quantum-classical based approach using variatonal quantum circuits for multi-class classification tasks on IRIS and MNIST datasets. Our study mainly contributes to Multi-class classification on the IRIS flower dataset with only four features and classification of MNIST digit on reduced dataset with three features. This paper is divided into several sections which are as follows, section 1 covers introduction. In section 2 we have covered related work on multi-class classification, and variational quantum circuits are explained in section 3. In section 4 we explained our proposed scheme and data preparation method, and section 5 is a discussion about results and conclusion 2. Related Work Quantum Neural Networks have been studied for the classification of classical data. It is the most explored method for classification with quantum computers. Wu Jindi et al. have proposed a novel quantum neural networks approach for classification [ 14 ]. It is a scalable approach in which small quantum hardware is used cooperatively. The whole image is divided into several parts, and feature extraction is done on each part using small quantum devices. Extracted local features are sent to the quantum device to perform prediction by combining all local features in parallel. Using the MNIST dataset, they conducted experiments and evaluated the outcomes for binary classification. The limitation in this approach is its scalability and efficiency for larger or more complex data sets. Also experiments are conducted for binary classification tasks, it is not known whether this approach can be used for multiclass classification on classical data. In another paper, Wu Jindi et al. [ 15 ] proposed a variational quantum multi-classifier based on correlation and measurement. They make use of a single readout qubit’s quantum information, using the same variational ansatz as binary classifiers. The readout state is then reconstructed from the measurement data using the quantum state tomography approach. Then, by examining the connection between classes, they employ the variational quantum clustering technique to identify the quantum labels for classes. The evaluation findings show that they attained enhanced performance with minimal quantum resources and a basic ansatz. The limitation in this method is that it depends on quantum state tomography to reconstruct the readout state, which can be resource-intensive in terms of both quantum circuits and classical computation for state reconstruction. Hanrui Wang et al. [ 16 ] presented on-chip parametrized quantum circuit training with parameter shift. They find that gradients obtained by parameter shift have low fidelity, which causes a decrease in training accuracy. To achieve this, they also suggest probabilistic gradient pruning, which identifies gradients with potentially significant errors. They performed experiments with the Quantum Neural Network datasets on five classification tasks. The findings show that on-chip training can classify images into two and four classes with approximately 90 and 60 percent accuracy, respectively. Overall, they achieve comparable on-chip training accuracy to noise-free simulation. The key limitations of this approach revolve around the sensitivity to noise, potential drawbacks of gradient pruning, limited scalability, and generalization, and the dependence on specific quantum hardware. These factors make it challenging to apply the method to more complex, real-world quantum machine learning problems. Chalumuri, A. et.al. in [ 17 ], used a hybrid model for the classification. They introduced a quantum multi-class classifier as a parametrized circuit. State preparation is done using a unitary operation on a single qubit. Three benchmark datasets were used for their quantum simulations: the Wireless Indoor Localization, Banknote Authentication (BNA), and the Iris dataset. The QMCC model identified the Iris, BNA, and WIL datasets with an accuracy of 92.10 percent, 89.50 percent, and 91.73 percent, respectively. The limitations are that the model might be prone to overfitting on small datasets, especially when using a parametrized quantum circuit with a large number of free parameters. Kevin Shen et al. improve a variational algorithm [ 18 ] that generally prepares the encoded data to solve the data encoding problem. The Fashion-MNIST dataset is encoded using the most recent technique. They provide a proof of concept for the near-term practicality of our data encoding technique by deploying basic quantum variational classifiers that are trained on the encoded dataset on a modern quantum computer and achieve moderate accuracy. The main limitations of the approach include the reliance on shallow circuits, which may not scale well to more complex problems, the challenges of data encoding, limited benchmarking on simple datasets, and optimization difficulties. 3. Variational Quantum Circuits In supervised learning, every machine learning model is trained on training data along with their labels. After learning from a dataset’s patterns, the algorithms create a model that can be used to predict the label of unknown data. The unpredictability characteristic of quantum mechanics is utilized while training to strengthen the model because machine learning strongly relies on linear algebra. They can be used to address various tasks, such as reinforcement learning, supervised learning, and unsupervised learning. Quantum computing concepts have enabled conventional randomized algorithms to perform exponentially faster than standard algorithms. Another technique for completing the QML task is variation quantum circuits [ 19 , 20 ]. The variational quantum circuit can be used as an artificial neural network, which is a hybrid approach utilizing quantum and classical computation power. It is also called a parameterized quantum circuit, where parameters are considered as weights for the neural network. These parameters are updated on the classical system in each epoch. We can use these quantum circuits to find the cost function, and it should be as simple as possible for a successful machine-learning model to work. We utilize a traditional computer to tune the parameters. This approach makes extensive use of parameterized, optimized quantum gates. Usually, these gates are a combination of rotation gates: R X , R Y , R Z and CNOT gates. The optimized circuit is used for classification. Figure 1 illustrates the overall view of variational quantum circuits with quantum and classical parts. Generally, VQC can be written as: U(θ) ψ = ∏ n i=1 U i ψ (1) where U(θ) is universal gates with as a parameter, n is the number of total gates, and ψ is input quantum states. By changing parameters, the operation of U can be modified. Every variational algorithm consist of the following steps Data encoding - A key component of VQC is data encoding, where classical data must be embedded into a quantum state before using it. This can be accomplished in several ways, with basis and amplitude encoding being the most popular methods. Ansatz design - This step involves the design of quantum circuits using quantum gates and making qubits into superposition and entanglement. Measurement - Apply the measurement to collapse the quantum state into either 0 or 1 binary state. Post-processing - It is the mapping of binary output obtained from VQC with the labels of the dataset for classification. We check here whether the predicted labels are correct or not. Optimization - In this step, the parameter optimization is done classically to reduce the cost function. We can use any classical optimizer to optimize the parameter, such as gradient descent, ADAM, or stochastic gradient descent. 4. Methodology A hybrid quantum-classical approach is used for multi-class classification using VQC. Where classical data or features will be encoded to quantum states, this step is also known as feature mapping. The quantum ansatz, which consists of entangling and rotational gates, receives these prepared states as an input. This ansatz is parameterized by angles that can be adjusted during training. The output of this circuit is measured to yield bit strings that represent classification results. For training the dataset will be divided into training and validation sets. The training set will consist of labeled data points, while the validation will be used to assess model performance after training. A classical optimization algorithm is employed to minimize a loss function, which compares predicted outputs with actual labels. The parameters (angles) of the quantum circuit are iteratively adjusted based on this optimization process. After processing through the quantum circuit, measurements are taken from the qubits. The resulting bit strings are interpreted as probabilities for class membership for classification tasks. 4.1 Data Preparation Let us explore our dataset. Mostly, data scientists are familiar with the Iris data. The data set comprises three classes: Virginica, Versicolor, and Setosa, each of which has 50 instances. The data have 150 instances. Four features are present in each instance: sepal length, sepal breadth, petal length, and petal width. As each class has an equal number of instances, the data set is properly balanced. Next, is the MNIST digit dataset of image 28 by 28 pixels. A total of 70000 images are there in the MNIST dataset, in which 60000 and 10,000 are used for training and testing purposes, respectively. To speed up the model training and the model’s performance, it might be beneficial in some circumstances to decrease the number of images from the dataset. We are selecting a reduced dataset for three classes as we cannot accommodate all features. Figure 2 and Fig. 3 are visualizations of data points of the selected data set 4.2 Data Encoding One of the most important and crucial steps in working with VQC is quantum data encoding for accelerated QML algorithms. 1) Basis Encoding: The simplest quantum encoding approach for arithmetic operations is the basis encoding technique. This method maps the binary form of classical data onto quantum base states. Typically, n qubits are needed for the binary representation of n classical data points using the basis encoding approach. The classical data point x = (x 1 , x 2 , ....x n ) will be encoded as ψ x =⊗ n i x i (2) 2) Amplitude Encoding: The amplitude encoding represents input data of x = (x 1 , x 2 , ....x n ) T of dimension N = 2 N as amplitudes of an n-qubit quantum state ϕ(x) as U ϕ (x) : x ∈ R N → ϕ(x) = 1 /x * Σ N i=1 x i i (3) where i is the ith computational basis state. 3) Angle Encoding: The classical data is loaded using the angle encoding method as the radians of rotation gates operating on qubits. To encode n data points we need n qubits and n rotation gates of R x , R y , R z acting on qubits. if x = (x 1 , x 2 , ....x n ) then states will be prepared as ψ x = ⊗ n i R(x i )x i (4) The worst-case time complexity of quantum encoding is exponential. LaRose et al. have presented different robust quantum encoding methods such as dense angle encoding, general qubit encoding, wavefunction encoding, and amplitude encoding and their results on different channels [ 22 ]. Angle coding and dense angle coding can be used to decrease a quantum circuit’s depth, and these two encoding techniques will need O(M) qubits to encode M-dimensional classical data. Whereas, the amplitude encoding will require O(logM) qubits, the depth of the circuit will be O(M). When the circuit depth is too deep, the quantum state may suffer from noise in the environment. We should select an encoding that can create a balance between a number of qubits and circuit depth. In our case of IRIS dataset for data encoding and state preparation we have used the standard ZZFeatureMap from the Qiskit library. The first step is to normalize the input features between 0 and 1. Each normalized feature is mapped to a rotational angle on the Bloch sphere using the AngleEmbedding technique. After applying the rotation gates, the entangling gates are used to create correlations between qubits, enhancing the model’s ability to capture complex relationships among features. To encode 4 features we have used 4 qubits. Each qubit has a rotation gate RZ that applies a rotation based on the feature value. Hence 4 rotations gates will be required and the rotation angle will be in the range between [0, 2π]. After the rotations, entanglement is applied between pairs of qubits using CZ gates. 6 CZ gates are used to make entanglement for all pairs of qubits. Total 10 gates are used in 1 repetition (reps). Here we have used 2 reps to provide a balance between accuracy and computational efficiency resulting in a circuit depth 20 that is often sufficient for achieving good classification performance without excessive complexity. Similar work has been done for reduced MNIST dataset. 4.3 Circuit Design and Training For the Iris flower dataset after the state preparation, we will design an ansatz for classification. The ansatz provides the parameterized structure needed to perform optimization via classical optimizers. These gates in the ansatz are parameterized, and the parameters of these gates are what need to be optimized during training to minimize classification error. Choosing the right ansatz is crucial for the performance of quantum models. The layers in classical neural networks are directly similar to this ansatz. Each gate in this circuit works as a node in the neural network. It has a set of adjustable weights. The gap between the predictions and the known labeled data is described by the cost function. To minimize a cost function, we need to optimize the weights. In Fig. 4 our circuit is plotted. There are 12 parameters in this circuit, ranging from 0 to 11. These are the trainable weights for the classifier. This circuit will act as the feed-forward layer of the neural network. The training includes learning the trainable parameters. In our case, learnable parameters are rotations, angles, entanglement operations, etc. of the quantum circuit. To minimize the loss function, which measures the difference between predicted and true labels, the loss function is computed using the measurements from the quantum circuit, and the optimizer updates the parameters iteratively. In this training process COBYLA optimizer is used for optimizing the parameters. COBYLA is designed for optimizing non-linear, non-differentiable functions. It is a gradient-free, derivative-free optimizer. Its simplicity and gradient-free nature make it a strong contender for quantum optimization tasks, especially those involving variational methods. The training time will increase if we select a gradient-based optimizer. However, for smooth, well-behaved problems, or problems that can efficiently provide gradients, other optimizers like L-BFGS-B, SPSA or ADAM might offer faster convergence and better performance. The stochastic optimizer such as Simultaneous Perturbation Stochastic Approximation (SPSA) can also be used for optimization. Now we have our features, ansatz, and optimizer ready, we can train our classifier. The hyperparameters that we have tuned to achieve optimal performance for training a variational quantum circuit. 4 qubits are used to encode 4 features, where 4 Rz rotation gates and 6 CX gates are applied to make qubits entangled, the circuit depth is 20 for feature mapping and 12 gates are used for creating ansatz for classification. To make the model converge faster we have used COBYLA optimizer with learning rate 0.001. During training the batch size is 16 and the number of epochs are 250. Trained on IBM Quantum simulator. We can train VQC using either a simulator or a real quantum computer. Here we will be using a quantum simulator as present real quantum hardware is noisy. Near the end of the 250 iterations of training, the cost function is not converging, we can see in Fig. 5 , indicating that the model’s performance will not change even after increasing the number of iterations. After training completion, we achieved a high score on the train set and test set. The test accuracy is 85 percent, whereas the training accuracy is 87 percent, and labels from the IRIS dataset of unseen data can be predicted using this model. In this situation, while designing the circuit, we have modified the reps parameter, which dictates how many times we add a quantum gate to the circuit, which is similar to adding a hidden layer in the classical neural network. A greater number of quantum gates results in more parameters and more entanglement operations. The model is therefore more flexible, but it also becomes more complex due to the increased number of parameters, and it typically takes longer to train. Even a minor change to the ansatz might produce improved outcomes; this indicates that the selection of hyperparameters is just as important in QML as it is in classical machine learning, and it could take some time to find the best values. As Iris data has only four features, we will be required to use only 4 qubits to process all the data, although this might not always be the case. If a data set contains more features than a modern quantum computer can accommodate, then we reduce the number of features, resulting in decreased performance for all models. Next we will train a model for the MNIST digit data set. Where the primary task is to identify which digit is present in the image ranging from 0 to 9. The size of each image of the MNIST data set is 28x28 pixels. Due to a limited number of qubits available, we cannot feed all features to the quantum circuit for training; hence, first of all, we will apply scaling to images to reduce the image size, and then we will reduce the dimensions to 3 features using Principal Component Analysis (PCA). The custom data is divided into an 80:10:10 ratio for training, validation, and testing, and we are using the categorical encoding of -1, 0, and 1 for class labels. Here we are using only three class classifications. Features are encoded using angle encoding with the Rx gate, whereas the output is the string of predicted labels based on the test dimension. The Ansatz used for the classification is given in Fig. 6 . Here we have used the IBM Qiskit simulator to build the circuit and train the VQC. From Fig. 7 we can see that the model is at minimum loss in 100 epochs. 5. Result and Discussion We have achieved an accuracy of 99 percent with classical Support Vector Machine (SVM) on the training data and 97 percent on the test data of the Iris data set, respectively, whereas we can achieve a test accuracy of 85 percent and training accuracy of 87 percent using VQC. As we can see, the classical support vector machine produced the best results. However, the four-feature-trained quantum model is also quite good for the IRIS dataset. Unsurprisingly, classical models outperform their quantum versions; nonetheless, classical ML has advanced significantly, whereas quantum ML has yet to achieve that degree of maturity. Three-class classification is done with the MNIST digit dataset with completely different circuits. In the MNIST dataset, classically we have achieved 99.82 percent accuracy on a training set using the Convolutional Neural Network (CNN). To perform our experiments, only classes 0, 1, and 2 are selected from the dataset. The same process is followed for training the variational quantum circuit (VQC) as training on the IRIS data set. Firstly, feature extraction is done with PCA, and these features are encoded into quantum states using a feature map; then these features are passed as input to the quantum circuit for training. Here we have trained our variational quantum circuit with a learning rate of 0.01, batch size 16, and Adam (Adaptive Moment Estimation) optimizer with 0.425 for 100 epochs. We can achieve nearly 97.95 percent accuracy on this reduced dataset with only three features Table 1 Experimental results of accuracy on different datasets. Data Set Classes Features Test Accuracy Train Accuracy Quantum Classical Quantum Classical IRIS 3 4 0.85 0.99 (With SVM) 0.87 0.97 (With SVM) MNIST 3 3 0.96 99.4 (With CNN) 0.97 99.82 (With CNN) We have successfully performed multi-class classification with minimum features on quantum hardware using a variational quantum circuit. In the case of the IRIS dataset, the number of features is four, whereas on the reduced MNIST digit dataset with only three features. We have used a single qubit for single features to train our model for addressing the constraints of the noisy intermediate-scale quantum (NISQ) era and to maximize computational efficiency. The single-qubit encoding mimics traditional convolutional neural network (CNN) techniques while significantly reducing the number of parameters compared to classical methods. This approach maintains spatial relationships between pixels, which are encoded into the single qubit. In Table 1 we have done comparative analysis of classical convolutional neural networks and VQC for multi-class classification using MNIST digit dataset We have seen that Shao et.al. [ 23 ] have achieved an training accuracy of 99.8 percent on this dataset after hyperparameter optimization. They have included all 28×28 = 784 features for classification while VQC has used only 3 three features to achieve comparable accuracy. As the number of available qubits will increase then we will be able to accommodate more features to surpass this achieved accuracy on classical computers for larger dimension data The complexity of quantum computations will scale with the size of the input image due to the increasing number of quantum operations required for encoding, processing, and decoding larger data. Larger images necessitate more qubits to represent pixel data, as each qubit typically encodes a specific part of the image (e.g., pixel intensity or feature). This increases the circuit depth and the complexity of state preparation. The circuit depth increases with the image size, as quantum circuits grow in size, the likelihood of errors increases. The fidelity of quantum gates tends to decrease as the circuit size increases. As quantum systems are inherently prone to errors due to noise, decoherence, and imperfect gate fidelity. Quantum models are promising but still in the early stages of development and face significant challenges in areas like data encoding, noise management, circuit complexity, and optimization. On simple tasks like the Iris dataset, classical models are likely to outperform quantum models due to their efficiency, maturity, and ability to handle smaller datasets effectively. Quantum models may show their potential in more complex, high-dimensional problems where their ability to leverage quantum states could offer advantages over classical models. Today, many quantum hardware and libraries are available to design quantum circuits and test and verify results on real hardware and simulators [ 24 ]. In this paper, we have used an IBM quantum simulator and hardware for evaluating our results. 6. Conclusion Many complex problems that are challenging for classical computers to resolve could be resolved by quantum computers. One of the most significant advantages of quantum computing is quantum parallelism. For example with n dimensions, a quantum computer can create a superposition of all 2n possible states simultaneously, while a classical computer would have to examine each state one by one. In high-dimensional data, training deep neural networks or other complex models involves minimizing loss functions over many parameters. Quantum optimization could speed up the convergence of these models by exploring the parameter space more efficiently. Quantum feature maps can map high-dimensional data into a quantum state that may require fewer qubits to represent, allowing quantum algorithms to process the data more efficiently. But current quantum computers are still in the NISQ era, which means we have a limited number of qubits and these qubits are noisy which restrict us to achieve quantum advantage. A shallow entangled circuit is designed to train VQC with the fewest possible trainable parameters. In this study, we demonstrated that quantum computing may be utilized to train neural networks and get results comparable to those of classical neural networks. Here, we demonstrated the use of variational quantum circuits for multi-class classification. Only a few parameters were used by VQC to identify the dataset’s complex and nonlinear patterns. It is therefore computationally efficient. With minor modifications, the ansatz employed in this paper can be applied to different classification tasks. The future scope of this work is the study of multiclass classification with a large data set of higher dimensions. Declarations Conflicts of Interest The authors declare that they have no conflict of interest. Author Contribution All the authors contributed equally to this work Data Availability The datasets generated and/or analysed during the current study are available in the following repository, IRIS data set : https://gist.github.com/curran/a08a1080b88344b0c8a7 MNIST data set: https://drive.google.com/file/d/1eEKzfmEu6WKdRlohBQiqi3PhW uIVJVP/view?usp = sharing References Shor, Peter W. ”Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.” SIAM review 41, no. 2 (1999): 303-332. 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Supplementary Files iris.csv mnisttest.csv Cite Share Download PDF Status: Posted Version 1 posted 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-6069218","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":440425019,"identity":"0ac1bf18-563d-483c-a992-47487e422abc","order_by":0,"name":"Muhammad 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Pal","email":"","orcid":"","institution":"University of Delhi","correspondingAuthor":false,"prefix":"","firstName":"Om","middleName":"Pal","lastName":"Pal","suffix":""}],"badges":[],"createdAt":"2025-02-20 06:53:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6069218/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6069218/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":80307187,"identity":"2f1b98ba-1931-4ea8-9e4b-4edef204011a","added_by":"auto","created_at":"2025-04-10 10:27:33","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":45722,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic illustration of variational quantum circuit for classification.\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/8859665385f84e91157523a1.jpg"},{"id":80307177,"identity":"7e24d5fe-e7b4-4021-99f0-204ac5bd2ae5","added_by":"auto","created_at":"2025-04-10 10:27:33","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":57003,"visible":true,"origin":"","legend":"\u003cp\u003eVisualization of Data point of IRIS flower Dataset\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/d05dd09e095b67bceb1729ab.jpg"},{"id":80307432,"identity":"2d57029c-1478-4754-b51a-cf1d68ca81e8","added_by":"auto","created_at":"2025-04-10 10:35:33","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":64715,"visible":true,"origin":"","legend":"\u003cp\u003eVisualization of Data point of Reduced MNIST digit Dataset\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/8ffdcb2755b39e2952897327.jpg"},{"id":80307166,"identity":"d8c7b5b9-66e4-4fc1-84f5-da934d9b009c","added_by":"auto","created_at":"2025-04-10 10:27:33","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":45717,"visible":true,"origin":"","legend":"\u003cp\u003eQuantum anstaz used for the classification of IRIS Dataset\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/d0c397b80cf5828d2f5630c7.jpg"},{"id":80307429,"identity":"efc418d2-80ab-45dd-b080-043b37d2f9ed","added_by":"auto","created_at":"2025-04-10 10:35:33","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":35788,"visible":true,"origin":"","legend":"\u003cp\u003eRepresentation of Loss decay during training after every iteration.\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/6500e9498980debae81c9948.jpg"},{"id":80307174,"identity":"6175cd8c-7920-4db4-9b76-607691638843","added_by":"auto","created_at":"2025-04-10 10:27:33","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":34591,"visible":true,"origin":"","legend":"\u003cp\u003eQuantum Ansatz used for the classification in MNIST Dataset.\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/83195cb4d2a21d0cbb2a0571.jpg"},{"id":80307169,"identity":"99349ce6-8217-44c3-965d-11c9413f0d18","added_by":"auto","created_at":"2025-04-10 10:27:33","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":29893,"visible":true,"origin":"","legend":"\u003cp\u003eRepresentation of Loss decay during training after every iteration.\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/ab84510d763a7c70601cb40c.jpg"},{"id":81579836,"identity":"613e829a-a6b6-4242-bc8e-80c81754fa22","added_by":"auto","created_at":"2025-04-28 18:46:28","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":784816,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/943e7b64-2902-443d-a3d9-d92fefc8cc5a.pdf"},{"id":80307168,"identity":"6aee5074-13ef-400b-b10f-78e7d794b8b9","added_by":"auto","created_at":"2025-04-10 10:27:33","extension":"csv","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":3975,"visible":true,"origin":"","legend":"","description":"","filename":"iris.csv","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/50ea508f3d9f99c49abed2e4.csv"},{"id":80307188,"identity":"693ef9d8-254d-424a-b748-36f6b01fb69f","added_by":"auto","created_at":"2025-04-10 10:27:33","extension":"csv","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":18289443,"visible":true,"origin":"","legend":"","description":"","filename":"mnisttest.csv","url":"https://assets-eu.researchsquare.com/files/rs-6069218/v1/e1f7a2104a4b80fb25930d6c.csv"}],"financialInterests":"No competing interests reported.","formattedTitle":"Multiclass Classification using VariationalQuantum Circuit on Benchmark Dataset","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eWe have seen massive growth in artificial intelligence in the past three decades. There are a wide range of applications of artificial intelligence in healthcare, agriculture, transportation, finance, the advertising industry, and many more. Generally, machine learning model training requires vast computational resources and time; sometimes training a deep learning model may take several days and even a month. Despite recent advances in classical computers, modern computers are reaching their limits in implementing machine learning in different areas. Researchers and computer scientists are looking for an alternative to classical computing, and they see quantum computers as an alternate solution to overcome this problem. The quantum computers have shown speedup in factorization of numbers [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] and searching in unstructured data [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Quantum computers use quantum phenomena called superposition, decoherence, and\u003c/p\u003e \u003cp\u003eentanglement to process quantum data.\u003c/p\u003e \u003cp\u003eQML is an emerging discipline, an intersection of quantum computing and artificial intelligence. Though QML is still in its initial stages, a variety of QML algorithms have still been proposed. QML has been studied for supervised, unsupervised, semi supervised, and reinforcement learning [\u003cspan additionalcitationids=\"CR4\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Today we have QSVM [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], quantum k nearest neighbor [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] for the binary classification task. Various studies have been done regarding supervised, semi-supervised, and unsupervised learning using quantum-classical-based machine learning algorithms [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. D.K. Park et al. have proposed a kernel-based quantum classifier [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. We also have witnessed the power of Hybrid-QCNN for multi-class classification tasks [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Adhikary et. al. [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] have used N level quantum systems to encode features for a Hybrid Quantum Classical classifier (HQC) for classification on different datasets. Davis Arthur and Prasanna Date[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] have proposed HQC neural network architecture for binary classification and [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] have demonstrated 3 class classification with high accuracy using amplitude encoding.\u003c/p\u003e \u003cp\u003eHere we present a mixed quantum-classical based approach using variatonal quantum circuits for multi-class classification tasks on IRIS and MNIST datasets. Our study mainly contributes to Multi-class classification on the IRIS flower dataset with only four features and classification of MNIST digit on reduced dataset with three features.\u003c/p\u003e \u003cp\u003eThis paper is divided into several sections which are as follows, section 1 covers introduction. In section 2 we have covered related work on multi-class classification, and variational quantum circuits are explained in section 3. In section 4 we explained our proposed scheme and data preparation method, and section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e5\u003c/span\u003e is a discussion about results and conclusion\u003c/p\u003e"},{"header":"2. Related Work","content":"\u003cp\u003eQuantum Neural Networks have been studied for the classification of classical data. It is the most explored method for classification with quantum computers. Wu Jindi et al. have proposed a novel quantum neural networks approach for classification [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. It is a scalable approach in which small quantum hardware is used cooperatively. The whole image is divided into several parts, and feature extraction is done on each part using small quantum devices. Extracted local features are sent to the quantum device to perform prediction by combining all local features in parallel. Using the MNIST dataset, they conducted experiments and evaluated the outcomes for binary classification. The limitation in this approach is its scalability and efficiency for larger or more complex data sets. Also experiments are conducted for binary classification tasks, it is not known whether this approach can be used for multiclass classification on classical data. In another paper, Wu Jindi et al. [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] proposed a variational quantum multi-classifier based on correlation and measurement. They make use of a single readout qubit\u0026rsquo;s quantum information, using the same variational ansatz as binary classifiers. The readout state is then reconstructed from the measurement data using the quantum state tomography approach. Then, by examining the connection between classes, they employ the variational quantum clustering technique to identify the quantum labels for classes. The evaluation findings show that they attained enhanced performance with minimal quantum resources and a basic ansatz. The limitation in this method is that it depends on quantum state tomography to reconstruct the readout state, which can be resource-intensive in terms of both quantum circuits and classical computation for state reconstruction.\u003c/p\u003e \u003cp\u003eHanrui Wang et al. [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] presented on-chip parametrized quantum circuit training with parameter shift. They find that gradients obtained by parameter shift have low fidelity, which causes a decrease in training accuracy. To achieve this, they also suggest probabilistic gradient pruning, which identifies gradients with potentially significant errors. They performed experiments with the Quantum Neural Network datasets on five classification tasks. The findings show that on-chip training can classify images into two and four classes with approximately 90 and 60 percent accuracy, respectively. Overall, they achieve comparable on-chip training accuracy to noise-free simulation. The key limitations of this approach revolve around the sensitivity to noise, potential drawbacks of gradient pruning, limited scalability, and generalization, and the dependence on specific quantum hardware. These factors make it challenging to apply the method to more complex, real-world quantum machine learning problems.\u003c/p\u003e \u003cp\u003eChalumuri, A. et.al. in [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], used a hybrid model for the classification. They introduced a quantum multi-class classifier as a parametrized circuit. State preparation is done using a unitary operation on a single qubit. Three benchmark datasets were used for their quantum simulations: the Wireless Indoor Localization, Banknote Authentication (BNA), and the Iris dataset. The QMCC model identified the Iris, BNA, and WIL datasets with an accuracy of 92.10 percent, 89.50 percent, and 91.73 percent, respectively. The limitations are that the model might be prone to overfitting on small datasets, especially when using a parametrized quantum circuit with a large number of free parameters. Kevin Shen et al. improve a variational algorithm [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] that generally\u003c/p\u003e \u003cp\u003eprepares the encoded data to solve the data encoding problem. The Fashion-MNIST dataset is encoded using the most recent technique. They provide a proof of concept for the near-term practicality of our data encoding technique by deploying basic quantum variational classifiers that are trained on the encoded dataset on a modern quantum computer and achieve moderate accuracy. The main limitations of the approach include the reliance on shallow circuits, which may not scale well to more complex problems, the challenges of data encoding, limited benchmarking on simple datasets, and optimization difficulties.\u003c/p\u003e"},{"header":"3. Variational Quantum Circuits","content":"\u003cp\u003eIn supervised learning, every machine learning model is trained on training data along with their labels. After learning from a dataset\u0026rsquo;s patterns, the algorithms create a model that can be used to predict the label of unknown data. The unpredictability characteristic of quantum mechanics is utilized while training to strengthen the model because machine learning strongly relies on linear algebra. They can be used to address various tasks, such as reinforcement learning, supervised learning, and unsupervised learning. Quantum computing concepts have enabled conventional randomized algorithms to perform exponentially faster than standard algorithms. Another technique for completing the QML task is variation quantum circuits [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe variational quantum circuit can be used as an artificial neural network, which is a hybrid approach utilizing quantum and classical computation power. It is also called a parameterized quantum circuit, where parameters are considered as weights for the neural network. These parameters are updated on the classical system in each epoch. We can use these quantum circuits to find the cost function, and it should be as simple as possible for a successful machine-learning model to work. We utilize a traditional computer to tune the parameters. This approach makes extensive use of parameterized, optimized quantum gates. Usually, these gates are a combination of rotation gates: R\u003csub\u003eX\u003c/sub\u003e, R\u003csub\u003eY\u003c/sub\u003e, R\u003csub\u003eZ\u003c/sub\u003e and CNOT gates. The optimized circuit is used for classification. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e illustrates the overall view of variational quantum circuits with\u003c/p\u003e \u003cp\u003equantum and classical parts. Generally, VQC can be written as:\u003c/p\u003e \u003cp\u003e \u003cem\u003eU(θ)\u003c/em\u003e ψ = \u0026prod;\u003csup\u003en\u003c/sup\u003e\u003csub\u003ei=1\u003c/sub\u003e \u003cem\u003eU\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003eψ (1)\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eU(θ)\u003c/em\u003e is universal gates with as a parameter, n is the number of total gates, and ψ is input quantum states. By changing parameters, the operation of U can be modified. Every variational algorithm consist of the following steps\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eData encoding - A key component of VQC is data encoding, where classical data must be embedded into a quantum state before using it. This can be accomplished in several ways, with basis and amplitude encoding being the most popular methods.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eAnsatz design - This step involves the design of quantum circuits using quantum gates and making qubits into superposition and entanglement.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eMeasurement - Apply the measurement to collapse the quantum state into either 0 or 1 binary state.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ePost-processing - It is the mapping of binary output obtained from VQC with the labels of the dataset for classification. We check here whether the predicted labels are correct or not.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eOptimization - In this step, the parameter optimization is done classically to reduce the cost function. We can use any classical optimizer to optimize the parameter, such as gradient descent, ADAM, or stochastic gradient descent.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"4. Methodology","content":"\u003cp\u003eA hybrid quantum-classical approach is used for multi-class classification using VQC. Where classical data or features will be encoded to quantum states, this step is also known as feature mapping. The quantum ansatz, which consists of entangling and rotational gates, receives these prepared states as an input. This ansatz is parameterized by angles that can be adjusted during training. The output of this circuit is measured to yield bit strings that represent classification results. For training the dataset will be divided into training and validation sets. The training set will consist of labeled data points, while the validation will be used to assess model performance after training. A classical optimization algorithm is employed to minimize a loss function, which compares predicted outputs with actual labels. The parameters (angles) of the quantum circuit are iteratively adjusted based on this optimization process. After processing through the quantum circuit, measurements are taken from the qubits. The resulting bit strings are interpreted as probabilities for class membership for classification tasks.\u003c/p\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Data Preparation\u003c/h2\u003e \u003cp\u003eLet us explore our dataset. Mostly, data scientists are familiar with the Iris data. The data set comprises three classes: Virginica, Versicolor, and Setosa, each of which has 50 instances. The data have 150 instances. Four features are present in each instance: sepal length, sepal breadth, petal length, and petal width. As each class has an equal number of instances, the data set is properly balanced. Next, is the MNIST digit dataset of image 28 by 28 pixels. A total of 70000 images are there in the MNIST dataset, in which 60000 and 10,000 are used for training and testing purposes, respectively. To speed up the model training and the model\u0026rsquo;s performance, it might be beneficial in some circumstances to decrease the number of images from the dataset. We are selecting a reduced dataset for three classes as we cannot accommodate all features. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e are visualizations of data points of the selected data set\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Data Encoding\u003c/h2\u003e \u003cp\u003eOne of the most important and crucial steps in working with VQC is quantum data encoding for accelerated QML algorithms.\u003c/p\u003e \u003cp\u003e1) Basis Encoding: The simplest quantum encoding approach for arithmetic operations is the basis encoding technique. This method maps the binary form of classical data onto quantum base states. Typically, n qubits are needed for the binary representation of n classical data points using the basis encoding approach. The classical data point x = (x\u003csub\u003e1\u003c/sub\u003e, x \u003csub\u003e2\u003c/sub\u003e, ....x\u003csub\u003en\u003c/sub\u003e ) will be encoded as\u003c/p\u003e \u003cp\u003eψ\u003csub\u003ex\u003c/sub\u003e =\u0026otimes;\u003csup\u003en\u003c/sup\u003e\u003csub\u003ei\u003c/sub\u003e x\u003csub\u003ei\u003c/sub\u003e (2)\u003c/p\u003e \u003cp\u003e2) Amplitude Encoding: The amplitude encoding represents input data of x = (x\u003csub\u003e1\u003c/sub\u003e, x \u003csub\u003e2\u003c/sub\u003e, ....x \u003csub\u003en\u003c/sub\u003e ) T of dimension N\u0026thinsp;=\u0026thinsp;2 \u003csup\u003eN\u003c/sup\u003e as amplitudes of an n-qubit quantum state ϕ(x) as\u003c/p\u003e\u003cp\u003e \u003cem\u003eU\u003c/em\u003e \u003csub\u003e\u003cem\u003eϕ\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(x)\u003c/em\u003e : \u003cem\u003ex\u003c/em\u003e \u0026isin; R \u003csup\u003eN\u003c/sup\u003e \u0026rarr; \u003cem\u003eϕ(x)\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003e1 /x\u003c/em\u003e * Σ\u003csup\u003eN\u003c/sup\u003e \u003csub\u003ei=1\u003c/sub\u003e \u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e \u003cem\u003ei\u003c/em\u003e (3)\u003c/p\u003e \u003cp\u003ewhere i is the ith computational basis state.\u003c/p\u003e \u003cp\u003e3) Angle Encoding: The classical data is loaded using the angle encoding method as the radians of rotation gates operating on qubits. To encode n data points we need n qubits and n rotation gates of R\u003csub\u003ex\u003c/sub\u003e, R\u003csub\u003ey\u003c/sub\u003e, R\u003csub\u003ez\u003c/sub\u003e acting on qubits. if x = (x\u003csub\u003e1\u003c/sub\u003e, x\u003csub\u003e2\u003c/sub\u003e, ....x\u003csub\u003en\u003c/sub\u003e ) then states will be prepared as\u003c/p\u003e \u003cp\u003eψ\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e = \u0026otimes;\u003csup\u003en\u003c/sup\u003e \u003csub\u003ei\u003c/sub\u003e \u003cem\u003eR(x\u003c/em\u003e \u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e)x\u003c/em\u003e \u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e (4)\u003c/p\u003e \u003cp\u003eThe worst-case time complexity of quantum encoding is exponential. LaRose et al. have presented different robust quantum encoding methods such as dense angle encoding, general qubit encoding, wavefunction encoding, and amplitude encoding and their results on different channels [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. Angle coding and dense angle coding can be used to decrease a quantum circuit\u0026rsquo;s depth, and these two encoding techniques will need O(M) qubits to encode M-dimensional classical data. Whereas, the amplitude encoding will require O(logM) qubits, the depth of the circuit will be O(M). When the circuit depth is too deep, the quantum state may suffer from noise in the environment. We should select an encoding that can create a balance between a number of qubits and circuit depth.\u003c/p\u003e \u003cp\u003eIn our case of IRIS dataset for data encoding and state preparation we have used the standard ZZFeatureMap from the Qiskit library. The first step is to normalize the input features between 0 and 1. Each normalized feature is mapped to a rotational angle on the Bloch sphere using the AngleEmbedding technique. After applying the rotation gates, the entangling gates are used to create correlations between qubits, enhancing the model\u0026rsquo;s ability to capture complex relationships among features. To encode 4 features we have used 4 qubits. Each qubit has a rotation gate RZ that applies a rotation based on the feature value. Hence 4 rotations gates will be required and the rotation angle will be in the range between [0, 2π]. After the rotations, entanglement is applied between pairs of qubits using CZ gates. 6 CZ gates are used to make entanglement for all pairs of qubits. Total 10 gates are used in 1 repetition (reps). Here we have used 2 reps to provide a balance between accuracy and computational efficiency resulting in a circuit depth 20 that is often sufficient for achieving good classification performance without excessive complexity. Similar work has been done for reduced MNIST dataset.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Circuit Design and Training\u003c/h2\u003e \u003cp\u003eFor the Iris flower dataset after the state preparation, we will design an ansatz for classification. The ansatz provides the parameterized structure needed to perform optimization via classical optimizers. These gates in the ansatz are parameterized, and the parameters of these gates are what need to be optimized during training to minimize classification error. Choosing the right ansatz is crucial for the performance of quantum models. The layers in classical neural networks are directly similar to this ansatz. Each gate in this circuit works as a node in the neural network. It has a set of adjustable weights. The gap between the predictions and the known labeled data is described by the cost function. To minimize a cost function, we need to optimize the weights. In Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e our circuit is plotted. There are 12 parameters in this circuit, ranging from 0 to 11. These are the trainable weights for the classifier. This circuit will act as the feed-forward layer of the neural network. The training includes learning the trainable parameters. In our case, learnable parameters are rotations, angles, entanglement operations, etc. of the quantum circuit. To minimize the loss function, which measures the difference between predicted and true labels, the loss function is computed using the measurements from the quantum circuit, and the optimizer updates the parameters iteratively. In this training process COBYLA optimizer is used for optimizing the parameters. COBYLA is designed for optimizing non-linear, non-differentiable functions. It is a gradient-free, derivative-free optimizer. Its simplicity and gradient-free nature make it a strong contender for quantum optimization tasks, especially those involving variational methods. The training time will increase if we select a gradient-based optimizer. However, for smooth, well-behaved problems, or problems that can efficiently provide gradients, other optimizers like L-BFGS-B, SPSA or ADAM might offer faster convergence and better performance. The stochastic optimizer such as Simultaneous Perturbation Stochastic Approximation (SPSA) can also be used for optimization.\u003c/p\u003e \u003cp\u003eNow we have our features, ansatz, and optimizer ready, we can train our classifier. The hyperparameters that we have tuned to achieve optimal performance for training a variational quantum circuit. 4 qubits are used to encode 4 features, where 4 Rz rotation gates and 6 CX gates are applied to make qubits entangled, the circuit depth is 20 for feature mapping and 12 gates are used for creating ansatz for classification. To make the model converge faster we have used COBYLA optimizer with learning rate 0.001. During training the batch size is 16 and the number of epochs are 250. Trained on IBM Quantum simulator. We can train VQC using either a simulator or a real quantum computer. Here we will be using a quantum simulator as present real quantum hardware is noisy. Near the end of the 250 iterations of training, the cost function is not converging, we can see in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, indicating that the model\u0026rsquo;s performance will not change even after increasing the number of iterations.\u003c/p\u003e \u003cp\u003eAfter training completion, we achieved a high score on the train set and test set. The test accuracy is 85 percent, whereas the training accuracy is 87 percent, and labels from the IRIS dataset of unseen data can be predicted using this model. In this situation, while designing the circuit, we have modified the reps parameter, which dictates how many times we add a quantum gate to the circuit, which is similar to adding a hidden layer in the classical neural network. A greater number of quantum gates results in more parameters and more entanglement operations. The model is therefore more flexible, but it also becomes more complex due to the increased number of parameters, and it typically takes longer to train. Even a minor change to the ansatz might produce improved outcomes; this indicates that the selection of hyperparameters is just as important in QML as it is in classical machine learning, and it could take some time to find the best values. As Iris data has only four features, we will be required to use only 4 qubits to process all the data, although this might not always be the case. If a data set contains more features than a modern quantum computer can accommodate, then we reduce the number of features, resulting in decreased performance for all models.\u003c/p\u003e \u003cp\u003eNext we will train a model for the MNIST digit data set. Where the primary task is to identify which digit is present in the image ranging from 0 to 9. The size of each image of the MNIST data set is 28x28 pixels. Due to a limited number of qubits available, we cannot feed all features to the quantum circuit for training; hence, first of all, we will apply scaling to images to reduce the image size, and then we will reduce the dimensions to 3 features using Principal Component Analysis (PCA). The custom data is divided into an 80:10:10 ratio for training, validation, and testing, and we are using the categorical encoding of -1, 0, and 1 for class labels. Here we are using only three class classifications. Features are encoded using angle encoding with the Rx gate, whereas the output is the string of predicted labels based on the test dimension. The Ansatz used for the classification is given in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Here we have used the IBM Qiskit simulator to build the circuit and train the VQC. From Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e we can see that the model is at minimum loss in 100 epochs.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5. Result and Discussion","content":"\u003cp\u003eWe have achieved an accuracy of 99 percent with classical Support Vector Machine (SVM) on the training data and 97 percent on the test data of the Iris data set, respectively, whereas we can achieve a test accuracy of 85 percent and training accuracy of 87 percent using VQC. As we can see, the classical support vector machine produced the best results. However, the four-feature-trained quantum model is also quite good for the IRIS dataset. Unsurprisingly, classical models outperform their quantum versions; nonetheless, classical ML has advanced significantly, whereas quantum ML has yet to achieve that degree of maturity.\u003c/p\u003e \u003cp\u003eThree-class classification is done with the MNIST digit dataset with completely different circuits. In the MNIST dataset, classically we have achieved 99.82 percent accuracy on a training set using the Convolutional Neural Network (CNN). To perform our experiments, only classes 0, 1, and 2 are selected from the dataset. The same process is followed for training the variational quantum circuit (VQC) as training on the IRIS data set. Firstly, feature extraction is done with PCA, and these features are encoded into quantum states using a feature map; then these features are passed as input to the quantum circuit for training. Here we have trained our variational quantum circuit with a learning rate of 0.01, batch size 16, and Adam (Adaptive Moment Estimation) optimizer with 0.425 for 100 epochs. We can achieve nearly 97.95 percent accuracy on this reduced dataset with only three features\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\u003eExperimental results of accuracy on different datasets.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" 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=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eData Set\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eClasses\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFeatures\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003eTest Accuracy\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e \u003cp\u003eTrain Accuracy\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eQuantum\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eClassical\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eQuantum\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eClassical\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIRIS\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.85\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.99 (With SVM)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.87\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.97 (With SVM)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMNIST\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.96\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e99.4 (With CNN)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e99.82 (With CNN)\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\u003eWe have successfully performed multi-class classification with minimum features on quantum hardware using a variational quantum circuit. In the case of the IRIS dataset, the number of features is four, whereas on the reduced MNIST digit dataset with only three features. We have used a single qubit for single features to train our model for addressing the constraints of the noisy intermediate-scale quantum (NISQ) era and to maximize computational efficiency. The single-qubit encoding mimics traditional convolutional neural network (CNN) techniques while significantly reducing the number of parameters compared to classical methods. This approach maintains spatial relationships between pixels, which are encoded into the single qubit. In Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e we have done comparative analysis of classical convolutional neural networks and VQC for multi-class classification using MNIST digit dataset We have seen that Shao et.al. [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] have achieved an training accuracy of 99.8 percent on this dataset after hyperparameter optimization. They have included all 28\u0026times;28\u0026thinsp;=\u0026thinsp;784 features for classification while VQC has used only 3 three features to achieve comparable accuracy. As the number of available qubits will increase then we will be able to accommodate more features to surpass this achieved accuracy on classical computers for larger dimension data\u003c/p\u003e \u003cp\u003eThe complexity of quantum computations will scale with the size of the input image due to the increasing number of quantum operations required for encoding, processing, and decoding larger data. Larger images necessitate more qubits to represent pixel data, as each qubit typically encodes a specific part of the image (e.g., pixel intensity or feature). This increases the circuit depth and the complexity of state preparation. The circuit depth increases with the image size, as quantum circuits grow in size, the likelihood of errors increases. The fidelity of quantum gates tends to decrease as the circuit size increases. As quantum systems are inherently prone to errors due to noise, decoherence, and imperfect gate fidelity. Quantum models are promising but still in the early stages of development and face significant challenges in areas like data encoding, noise management, circuit complexity, and optimization. On simple tasks like the Iris dataset, classical models are likely to outperform quantum models due to their efficiency, maturity, and ability to handle smaller datasets effectively. Quantum models may show their potential in more complex, high-dimensional problems where their ability to leverage quantum states could offer advantages over classical models. Today, many quantum hardware and libraries are available to design quantum circuits and test and verify results on real hardware and simulators [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. In this paper, we have used an IBM quantum simulator and hardware for evaluating our results.\u003c/p\u003e"},{"header":"6. Conclusion","content":"\u003cp\u003eMany complex problems that are challenging for classical computers to resolve could be resolved by quantum computers. One of the most significant advantages of quantum computing is quantum parallelism. For example with n dimensions, a quantum computer can create a superposition of all 2n possible states simultaneously, while a classical computer would have to examine each state one by one. In high-dimensional data, training deep neural networks or other complex models involves minimizing loss functions over many parameters. Quantum optimization could speed up the convergence of these models by exploring the parameter space more efficiently. Quantum feature maps can map high-dimensional data into a quantum state that may require fewer qubits to represent, allowing quantum algorithms to process the data more efficiently. But current quantum computers are still in the NISQ era, which means we have a limited number of qubits and these qubits are noisy which restrict us to achieve quantum advantage.\u003c/p\u003e \u003cp\u003eA shallow entangled circuit is designed to train VQC with the fewest possible trainable parameters. In this study, we demonstrated that quantum computing may be utilized to train neural networks and get results comparable to those of classical neural networks. Here, we demonstrated the use of variational quantum circuits for multi-class classification. Only a few parameters were used by VQC to identify the dataset\u0026rsquo;s complex and nonlinear patterns. It is therefore computationally efficient. With minor modifications, the ansatz employed in this paper can be applied to different classification tasks. The future scope of this work is the study of multiclass classification with a large data set of higher dimensions.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eConflicts of Interest\u003c/h2\u003e \u003cp\u003eThe authors declare that they have no conflict of interest.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAll the authors contributed equally to this work\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e \u003cp\u003eThe datasets generated and/or analysed during the current study are available in the following repository,\u003c/p\u003e \u003cp\u003eIRIS data set : \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://gist.github.com/curran/a08a1080b88344b0c8a7\u003c/span\u003e\u003cspan address=\"https://gist.github.com/curran/a08a1080b88344b0c8a7\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003eMNIST data set: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://drive.google.com/file/d/1eEKzfmEu6WKdRlohBQiqi3PhW\u003c/span\u003e\u003cspan address=\"https://drive.google.com/file/d/1eEKzfmEu6WKdRlohBQiqi3PhW\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003euIVJVP/view?usp\u0026thinsp;=\u0026thinsp;sharing\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eShor, Peter W. \u0026rdquo;Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.\u0026rdquo; SIAM review 41, no. 2 (1999): 303-332.\u003c/li\u003e\n\u003cli\u003eGrover, Lov K. \u0026rdquo;A fast quantum mechanical algorithm for database search.\u0026rdquo; In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pp. 212-219. 1996.\u003c/li\u003e\n\u003cli\u003eKulkarni, Viraj, Milind Kulkarni, and Aniruddha Pant. \u0026rdquo;Quantum computing methods for supervised learning.\u0026rdquo; Quantum Machine Intelligence 3, no. 2 (2021):23.\u003c/li\u003e\n\u003cli\u003eOtterbach, Johannes S., Riccardo Manenti, Nasser Alidoust, A. 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Fedorov. \u0026rdquo;Multiclass classification using quantum convolutional neural networks with hybrid quantum-classical learning.\u0026rdquo; Frontiers in Physics 10 (2022): 1173.\u003c/li\u003e\n\u003cli\u003eAdhikary, Soumik, Siddharth Dangwal, and Debanjan Bhowmik. \u0026rdquo;Supervised learning with a quantum classifier using multi-level systems.\u0026rdquo; Quantum Information Processing 19 (2020): 1-12.\u003c/li\u003e\n\u003cli\u003eDate, Prasanna, Davis Arthur, and Lauren Pusey-Nazzaro. \u0026rdquo;QUBO formulations for training machine learning models.\u0026rdquo; Scientific reports 11, no. 1 (2021): 10029.\u003c/li\u003e\n\u003cli\u003eZhang, Anqi, Xiaoyun He, and Shengmei Zhao. \u0026rdquo;Quantum algorithm for neural network enhanced multi-class parallel classification.\u0026rdquo; arXiv preprint arXiv:2203.04097 (2022).\u003c/li\u003e\n\u003cli\u003eWu, Jindi, Zeyi Tao, and Qun Li. \u0026rdquo;wpScalable Quantum Neural Networks for Classification.\u0026rdquo; In 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), pp. 38-48. 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(2025). Comparative Study of Quantum Computing Tools and Frameworks. In: Malhotra, M. (eds) Innovation and Emerging Trends in Computing and Information Technologies. IETCIT 2024. Communications in Computer and Information Science, vol 2126. Springer, Cham. https://doi.org/10.1007/978-3-031-80842-58\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Neural Network, Multi-class Classification, Variational Quantum Circuits, Hybrid Quantum Classical Algorithm","lastPublishedDoi":"10.21203/rs.3.rs-6069218/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6069218/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eToday, classification is a significant task in data science, and many industries, including healthcare, transport, and banking sectors, are required to classify the data. In this NISQ era, quantum computers are capable of solving complex data challenges and can predict results with minimum features. The quantum neural network is being studied extensively for machine learning problems. In this paper, we have performed the multiclass classification using variational quantum circuits on benchmark datasets. A combination of quantum and classical neural networks is used to build the quantum circuit and optimize the parameters. The quantum circuit is used for the feedforward architecture, while in back-propagation, parameters are updated using a classical optimizer on classical computers. We have successfully shown classification using the proposed approach in benchmark datasets, such as the Iris flower and the MNIST Digit data set. Our results show that VQC is a promising candidate for classification problems with fewer features. To perform our experiments we have used IBM Quantum hardware and simulators.\u003c/p\u003e","manuscriptTitle":"Multiclass Classification using VariationalQuantum Circuit on Benchmark Dataset","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-04-10 10:27:27","doi":"10.21203/rs.3.rs-6069218/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"46296a51-9ecc-46fd-b03e-e87b7fbd44fb","owner":[],"postedDate":"April 10th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":46887587,"name":"Physical sciences/Engineering"},{"id":46887588,"name":"Physical sciences/Mathematics and computing"}],"tags":[],"updatedAt":"2025-04-28T18:38:21+00:00","versionOfRecord":[],"versionCreatedAt":"2025-04-10 10:27:27","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6069218","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6069218","identity":"rs-6069218","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}