Abstract We propose a model for data classification using isolated quantum $$\varvec{d}$$ d -level systems or else qudits. The procedure consists of an encoding phase where classical data are mapped on the surface of the qudit’s Bloch hyper-sphere via rotation encoding, followed by a rotation of the sphere and a projective measurement. The rotation is adjustable in order to control the operator to be measured, while additional weights are introduced in the encoding phase adjusting the mapping on the Bloch’s hyper-surface. During the training phase, a cost function based on the average expectation value of the observable is minimized using gradient descent thereby adjusting the weights. Using examples and performing a numerical estimation of lossless memory dimension, we demonstrate that this geometrically inspired qudit model for classification is able to solve nonlinear classification problems using a small number of parameters only and without requiring entangling operations.
Abstract We propose a model for data classification using isolated quantum \varvecd d -level systems or else qudits. The procedure consists of an encoding phase where classical data are mapped on the surface of the qudit’s Bloch hyper-sphere via rotation encoding, followed by a rotation of the sphere and a projective measurement. The rotation is adjustable in order to control the operator to be measured, while additional weights are introduced in the encoding phase adjusting the mapping on the Bloch’s hyper-surface. During the training phase, a cost function based on the average expectation value of the observable is minimized using gradient descent thereby adjusting the weights. Using examples and performing a numerical estimation of lossless memory dimension, we demonstrate that this geometrically inspired qudit model for classification is able to solve nonlinear classification problems using a small number of parameters only and without requiring entangling operations.
Abstract Quantum two-level systems, i.e., qubits, form the basis for most quantum machine learning approaches that have been proposed throughout the years. However, higher dimensional quantum systems constitute a promising alternative and are increasingly explored in theory and practice. Here, we explore the capabilities of multi-level quantum systems, so-called qudits, for their use in a quantum machine learning context. We formulate classification and regression problems with the data re-uploading approach and demonstrate that a quantum circuit operating on a single qudit is able to successfully learn highly non-linear decision boundaries of classification problems such as the MNIST digit recognition problem. We demonstrate that the performance strongly depends on the relation between the qudit states representing the labels and the structure of labels in the training data set. Such a bias can lead to substantial performance improvement over qubit-based circuits in cases where the labels, the qudit states, and the operators employed to encode the data are well-aligned. Furthermore, we elucidate the influence of the choice of the elementary operators and show that a squeezing operator is necessary to achieve good performances. We also show that there exists a trade-off for qudit systems between the number of circuit-generating operators in each processing layer and the total number of layers needed to achieve a given accuracy. Finally, we compare classification results from numerically exact simulations and their equivalent implementation on actual IBM quantum hardware. The findings of our work support the notion that qudit-based algorithms exhibit attractive traits and constitute a promising route to increasing the computational capabilities of quantum machine learning approaches.
This research explores the integration of quantum computing with classical machine learning to enhance data classification tasks using Quantum Neural Networks (QNN) and Parameterized Quantum Circuits (PQC). The hybrid approach leverages the advantages of both quantum and classical systems to improve the efficiency and accuracy of data processing. In this model, data is encoded into qubits using amplitude encoding, representing input vectors as amplitudes of quantum states. The QNN is initialized by placing the qubits in superposition using Hadamard gates, followed by data encoding with parameterized rotational gates that map classical data to quantum states using rotation angles. PQC plays a central role by applying layers of parameterized quantum operations to process data in the quantum space. These parameters are optimized during the training process, where a quadratic loss function minimizes the error between the predicted quantum states and the true class labels using gradient descent. Experiments conducted on the MNIST dataset show that the hybrid the hybrid quantum-classical neural network (QCNN) with PQC achieves a classification accuracy of over 95%, highlighting its potential in machine learning applications. The results demonstrate that integrating quantum computing with classical machine learning enhances performance in complex data analysis tasks due to the exponential growth of quantum state space and the parallelism of quantum systems, making hybrid models promising for computer vision and classification tasks.
Abstract We study an analogous Bloch sphere representation of higher-level quantum systems using the Heisenberg-Weyl operator basis. We introduce a parametrization method that will allow us to identify a real-valued Bloch vector for an arbitrary density operator. Before going into arbitrary d-level (d ≥ 3) quantum systems (qudits), we start our analysis with three-level ones (qutrits). It is well known that we need at least eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits). However, using our method we can divide these parameters into four weight, and four angular parameters, and find that the weight parameters are inducing a unit sphere in four-dimension. And, the four angular parameters determine whether a Bloch vector is physical. Therefore, unlike its qubit counterpart, the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this construction allows us to define different properties of qutrits in terms of Bloch vector components. We also examine the two and three-dimensional sections of the sphere, which reveal a non-convex yet closed structure for physical qutrit states. Further, we apply our representation to derive mutually unbiased bases (MUBs), characterize unital maps for qutrits, and assess ensembles using the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to qudits, showcasing its potential applicability beyond the qutrit scenario.
Abstract We present a comprehensive investigation into the learning capabilities of a simple d-level system (qudit). Our study is specialized for classification tasks using real-world databases, specifically the Iris, breast cancer, and MNIST datasets. We explore various learning models in the metric learning framework, along with different encoding strategies. In particular, we employ data re-uploading techniques and maximally orthogonal states to accommodate input data within low-dimensional systems. Our findings reveal optimal strategies, indicating that when the dimension of input feature data and the number of classes are not significantly larger than the qudit’s dimension, our results show favorable comparisons against the best classical models. This trend holds true even for small quantum systems, with dimensions d < 5 and utilizing algorithms with a few layers ( L = 1 , 2 ). However, for high-dimensional data such as MNIST, we adopt a hybrid approach involving dimensional reduction through a convolutional neural network. In this context, we observe that small quantum systems often act as bottlenecks, resulting in lower accuracy compared to their classical counterparts.
Abstract Machine learning can make predictions on unseen data by extracting information and constructing algorithms and statistical models from the training data, which is highly desirable for speech recognition and computer vision. Quantum computing harnesses the phenomena of quantum mechanics realizing parallel computing, which could solve certain computational problems, such as RSA encryption significantly faster than classical computers. Quantum machine learning is reasonable to enable machine learning faster than that of classical computers by utilizing quantum operations. Binary classification is a typical quantum machine learning task applied to single qubits. Here we present a binary classifier for regions. Firstly, we sample two sets of quantum data points located in the X-Z plane of the Bloch sphere randomly. Then we utilize a single parameterized rotation gate and measure the qubits along the Z axis. We train the hybrid model combined by quantum mdoel and classical Neutral network and regard the corss entropy the predictions of the classical Netural network and labels as the loss function. Based on the above, we present the results of the trained hybrid model to classify new quantum qubits.
Understanding the impact of small quantum gate perturbations, which are common in quantum digital devices but absent in classical computers, is crucial for identifying potential advantages in quantum machine learning. While these perturbations are typically seen as detrimental to quantum computation, they can actually enhance performance by serving as a natural source of data augmentation. Additionally, they can often be efficiently simulated on classical hardware, enabling quantum-inspired approaches to improve classical machine learning methods. In this paper, we investigate random Bloch sphere rotations, which are fundamental SU(2) transformations, as a simple yet effective quantum-inspired data augmentation technique. Unlike conventional augmentations such as flipping, rotating, or cropping, quantum transformations lack intuitive spatial interpretations, making their application to tasks like image classification less straightforward. While common quantum augmentation methods rely on applying quantum models or trainable quanvolutional layers to classical datasets, we focus on the direct application of small-angle Bloch rotations and their effect on classical data. Using the large-scale ImageNet dataset, we demonstrate that our quantum-inspired augmentation method improves image classification performance, increasing Top-1 accuracy by 3%, Top-5 accuracy by 2.5%, and the F1 score from 8% to 12% compared to standard classical augmentation methods. Finally, we examine the use of stronger unitary augmentations. Although these transformations preserve information in principle, they result in visually unrecognizable images with potential applications for privacy computations. However, we show that our augmentation approach and simple SU(2) transformations do not enhance differential privacy and discuss the implications of this limitation.