This article presents a quantum computing approach to designing of similarity measures and kernels for classification of stochastic symbolic time series. In the area of machine learning, kernels are important components of various similarity-based classification, clustering, and regression algorithms. An effective strategy for devising problem-specific kernels is leveraging existing generative models of the example space. In this study we assume that a quantum generative model, known as quantum hidden Markov model (QHMM), describes the underlying distributions of the examples. The sequence structure and probability are determined by transitions within model's density operator space. Consequently, the QHMM defines a mapping from the example space into the broader quantum space of density operators. Sequence similarity is evaluated using divergence measures such as trace and Bures distances between quantum states. We conducted extensive simulations to explore the relationship between the distribution of kernel-estimated similarity and the dimensionality of the QHMMs Hilbert space. As anticipated, a higher dimension of the Hilbert space corresponds to greater sequence distances and a more distinct separation of the examples. To empirically evaluate the performance of the kernels, we defined classification tasks based on a simplified generative model of directional price movement in the stock market. We implemented two widely-used kernel-based algorithms - support vector machines and k-nearest neighbors - using both classical and quantum kernels. Across all classification task scenarios, the quantum kernels consistently demonstrated superior performance compared to their classical counterparts.
This article presents a quantum computing approach to designing of similarity measures and kernels for classification of stochastic symbolic time series. In the area of machine learning, kernels are important components of various similarity-based classification, clustering, and regression algorithms. An effective strategy for devising problem-specific kernels is leveraging existing generative models of the example space. In this study we assume that a quantum generative model, known as quantum hidden Markov model (QHMM), describes the underlying distributions of the examples. The sequence structure and probability are determined by transitions within model's density operator space. Consequently, the QHMM defines a mapping from the example space into the broader quantum space of density operators. Sequence similarity is evaluated using divergence measures such as trace and Bures distances between quantum states. We conducted extensive simulations to explore the relationship between the distribution of kernel-estimated similarity and the dimensionality of the QHMMs Hilbert space. As anticipated, a higher dimension of the Hilbert space corresponds to greater sequence distances and a more distinct separation of the examples. To empirically evaluate the performance of the kernels, we defined classification tasks based on a simplified generative model of directional price movement in the stock market. We implemented two widely-used kernel-based algorithms - support vector machines and k-nearest neighbors - using both classical and quantum kernels. Across all classification task scenarios, the quantum kernels consistently demonstrated superior performance compared to their classical counterparts.
Support Vector Machines (SVMs) are powerful machine learning models widely used for classification and regression tasks. However, their performance can be significantly impacted by the choice of the kernel function, which defines how data points are mapped into a higher-dimensional space. Traditional kernels often struggle to capture complex, non-linear relationships effectively, leading to suboptimal results. This paper proposes a novel approach to kernel design inspired by quantum mechanical principles. Specifically, we leverage concepts of quantum entanglement and superposition to generate kernels that can potentially capture richer and more nuanced data patterns. Our core claim is that employing these quantum-inspired designs can improve SVM performance and scalability. We present a theoretical framework outlining how quantum mechanical effects can be translated into kernel functions. The proposed method is based on constructing kernel functions that implicitly encode correlations between data points, mimicking the inherent interconnectedness observed in quantum systems. We explore specific kernel formulations based on these principles and discuss their potential advantages and challenges. The ultimate goal is to demonstrate that quantum inspiration can unlock new avenues for kernel design, leading to more effective SVM models.
The quantum support vector machines (QSVMs) offer a paradigm change that may leverage quantum mechanical processes to outperform traditional support vector machines (SVMs) in high-dimensional and nonlinear classification tasks. These machines are located at the crossroads of supervised machine learning and quantum computing. The purpose of this chapter is to provide a comprehensive technical analysis of QSVMs, with a particular emphasis on the use of variation quantum circuits for implicit feature space transformations and the development of these circuits within the context of quantum kernel estimation. Through the use of quantum state preparation and interference, we provide a comprehensive explanation of the possible fabrication of quantum kernels that are based on inner product evaluations in exponentially huge Hilbert spaces. When it comes to boosting model expressivity beyond classical limits, the importance that entanglement and quantum feature mapping play is specifically stressed. In addition to this, we investigate algorithmic implementations on noisy intermediate-scale quantum (NISQ) hardware, focusing on problems associated with quantum decoherence, gate fidelity, and loop depth optimization. Several fields, including quantum-enhanced drug discovery, anomaly detection in quantum sensor networks, fault classification in quantum error correction, and portfolio optimization, are now the focus of research that is being conducted in order to develop real-world applications. This chapter comes to a close with an analysis of the existing theoretical limits, resource needs, and future perspectives for the use of QSVMs in machine learning applications in order to achieve a quantum advantage. In order to facilitate the development of scalable quantum machine learning models that are founded on strong mathematical and computational notions, this synthesis is geared at academics and practitioners.
Quantum computing offers a powerful new paradigm for enhancing classical machine learning, particularly in classification tasks. In this paper, we examine the use of quantum kernels within Support Vector Classification (SVC) and compare their performance to classical SVC. Our experiments focus on the Iris and Wine datasets. For the Wine dataset, we applied Principal Component Analysis (PCA), reducing the feature space from 13 to 4 dimensions to improve computational efficiency. The results demonstrate that Quantum-enhanced SVC (QSVC) outperforms classical SVC, achieving an accuracy improvement from 96.15% with classical SVM to 98.05% with QSVC for the Wine dataset. Similarly, for the Iris dataset, classical SVC achieves near-perfect accuracy at 100%, while QSVC closely follows with 97.25% accuracy. These results highlight the capability of quantum kernels to improve both accuracy and efficiency in classification tasks, providing a promising direction for scalable quantum-classical hybrid systems in machine learning.
The support vector machine algorithm with a quantum kernel estimator (QSVM-Kernel), as a leading example of a quantum machine learning technique, has undergone significant advancements. Nevertheless, its integration with classical data presents unique challenges. While quantum computers primarily interact with data in quantum states, embedding classical data into quantum states using feature mapping techniques is essential for leveraging quantum algorithms Despite the recognized importance of feature mapping, its specific impact on data classification outcomes remains largely unexplored. This study addresses this gap by comprehensively assessing the effects of various feature mapping methods on classification results, taking medical data analysis as a case study. In this study, the QSVM-Kernel method was applied to classification problems in two different and publicly available medical datasets, namely, the Wisconsin Breast Cancer (original) and The Cancer Genome Atlas (TCGA) Glioma datasets. In the QSVM-Kernel algorithm, quantum kernel matrices obtained from 9 different quantum feature maps were used. Thus, the effects of these quantum feature maps on the classification results of the QSVM-Kernel algorithm were examined in terms of both classifier performance and total execution time. As a result, in the Wisconsin Breast Cancer (original) and TCGA Glioma datasets, when Rx and Ry rotational gates were used, respectively, as feature maps in the QSVM-Kernel algorithm, the best classification performances were achieved both in terms of classification performance and total execution time. The contributions of this study are that (1) it highlights the significant impact of feature mapping techniques on medical data classification outcomes using the QSVM-Kernel algorithm, and (2) it also guides undertaking research for improved QSVM classification performance.
As quantum computers become increasingly practical, so does the prospect of using quantum computation to improve upon traditional algorithms. Kernel methods in machine learning is one area where such improvements could be realized in the near future. Paired with kernel methods like support-vector machines, small and noisy quantum computers can evaluate classically-hard quantum kernels that capture unique notions of similarity in data. Taking inspiration from techniques in classical machine learning, this work investigates simulated quantum kernels in the context of multiple kernel learning (MKL). We consider pairwise combinations of several classical-classical, quantum-quantum, and quantum-classical kernels in an empirical investigation of their classification performance with support-vector machines. We also introduce a novel approach, which we call QCC-net (quantum-classical-convex neural network), for optimizing the weights of base kernels together with any kernel parameters. We show this approach to be effective for enhancing various performance metrics in an MKL setting. Looking at data with an increasing number of features (up to 13 dimensions), we find parameter training to be important for successfully weighting kernels in some combinations. Using the optimal kernel weights as indicators of relative utility, we find growing contributions from trainable quantum kernels in quantum-classical kernel combinations as the number of features increases. We observe the opposite trend for combinations containing simpler, non-parametric quantum kernels.
Support vector machines are powerful and frequently used machine learning methods for classification and regression tasks, which rely on the construction of kernel matrices. While crucial for the performance of this machine learning approach, choosing the most suitable kernel is highly problem-dependent. The emergence of quantum computers and quantum machine learning techniques provides new possibilities for generating powerful quantum kernels. Within this work, we solve a real-world price forecasting problem using fidelity and projected quantum kernels, which are promising candidates for the utility of near-term quantum computing. In our analysis, we examine and validate the most auspicious quantum kernels from literature and compare their performance with an optimized classical kernel. Unlike previous work on quantum support vector machines, our dataset includes categorical features that need to be encoded as numerical features, which we realize by using the one-hot-encoding scheme. One-hot-encoding, however, increases the dimensionality of the dataset significantly, which collides with the current limitations of noisy intermediate scale quantum computers. To overcome these limitations, we use autoencoders to learn a low-dimensional representation of the feature space that still maintains the most important information of the original data. To examine the impact of autoencoding, we compare the results of the encoded date with the results of the original, unencoded dataset. We could demonstrate that quantum kernels are comparable to or even better than the classical support vector machine kernels regarding the mean absolute percentage error scores for both encoded and unencoded datasets.