Abstract This paper narrows the gap between previous literature on quantum linear algebra and practical data analysis on a quantum computer, formalizing quantum procedures that speed-up the solution of eigenproblems for data representations in machine learning. The power and practical use of these subroutines is shown through new quantum algorithms, sublinear in the input matrix’s size, for principal component analysis, correspondence analysis, and latent semantic analysis. We provide a theoretical analysis of the run-time and prove tight bounds on the randomized algorithms’ error. We run experiments on multiple datasets, simulating PCA’s dimensionality reduction for image classification with the novel routines. The results show that the run-time parameters that do not depend on the input’s size are reasonable and that the error on the computed model is small, allowing for competitive classification performances.
Abstract This paper narrows the gap between previous literature on quantum linear algebra and practical data analysis on a quantum computer, formalizing quantum procedures that speed-up the solution of eigenproblems for data representations in machine learning. The power and practical use of these subroutines is shown through new quantum algorithms, sublinear in the input matrix’s size, for principal component analysis, correspondence analysis, and latent semantic analysis. We provide a theoretical analysis of the run-time and prove tight bounds on the randomized algorithms’ error. We run experiments on multiple datasets, simulating PCA’s dimensionality reduction for image classification with the novel routines. The results show that the run-time parameters that do not depend on the input’s size are reasonable and that the error on the computed model is small, allowing for competitive classification performances.
This paper evaluates the performance of two quantum machine learning algorithms: Quantum Support Vector Machine (QSVM) and Quantum Principal Component Analysis (Q-PCA). The scalability of QSVM is analyzed across increasing dataset sizes and feature dimensions, revealing faster-than-linear growth in kernel evaluation and SVM training times. However, this computational overhead is accompanied by a notable decline in classification accuracy, highlighting challenges in applying QSVM to larger or more complex datasets. In contrast, Q-PCA demonstrates near-linear scalability with dataset size and qubit count, leveraging the independence of quantum circuit execution and structured operations to maintain efficiency. A QNN-based implementation of Q-PCA further validates its ability to align data with principal components, maximize variance, and enable effective dimensionality reduction. These findings emphasize the potential of Q-PCA for efficient quantum-enhanced data analysis while underscoring the need for optimization in QSVM and further exploration of Q-PCA’s capabilities for real-world quantum computing applications.
Abstract Kernel principal component analysis (kernel PCA) is a nonlinear dimensionality reduction technique that employs kernel functions to map data into a high-dimensional feature space, thereby extending the applicability of linear PCA to nonlinear data and facilitating the extraction of informative principal components. However, kernel PCA necessitates the manipulation of large-scale matrices, leading to high computational complexity and posing challenges for efficient implementation in big data environments. Quantum computing has recently been integrated with kernel methods in machine learning, enabling effective analysis of input data within intractable feature spaces. Although existing quantum kernel PCA proposals promise exponential speedups, they impose stringent requirements on quantum hardware that are challenging to fulfill. In this work, we propose a quantum algorithm for kernel PCA by establishing a connection between quantum kernel methods and block encoding, thereby diagonalizing the centralized kernel matrix on a quantum computer. The query complexity is logarithmic with respect to the size of the data vector, D , and linear with respect to the size of the dataset. An exponential speedup could be achieved when the dataset consists of a few high-dimensional vectors, wherein the dataset size is polynomial in log ( D ) , with D being significantly large. In contrast to existing work, our algorithm enhances the efficiency of quantum kernel PCA and reduces the requirements for quantum hardware. Furthermore, we have also demonstrated that the algorithm based on block encoding matches the lower bound of query complexity, indicating that our algorithm is nearly optimal. Our research has laid down new pathways for developing quantum machine learning algorithms aimed at addressing tangible real-world problems and demonstrating quantum advantages within machine learning.
We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given nd-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training classifiers with constant margin runs in O~(n+d) time. We design sublinear quantum algorithms for the same task running in O~(n+d) time, a quadratic improvement in both n and d. Moreover, our algorithms use the standard quantization of the classical input and generate the same classical output, suggesting minimal overheads when used as subroutines for end-to-end applications. We also demonstrate a tight lower bound (up to poly-log factors) and discuss the possibility of implementation on near-term quantum machines. As a side result, we also give sublinear quantum algorithms for approximating the equilibria of n-dimensional matrix zero-sum games with optimal complexity Θ~(n).
This essential resource bridges the gap between classical data limitations and the future of computing, giving you the scalable, quantum-accelerated PCA strategies needed to conquer today’s massive, high-dimensional datasets. With the rapid growth of big data in fields such as genomics, internet traffic analysis, and social network data, traditional principal component analysis methods have reached their limits in terms of scalability and computational efficiency. This volume delves into cutting-edge advancements in principal component analysis (PCA), particularly focusing on its applications in handling high-dimensional and large-scale datasets. It also provides practical insights into how PCA can be applied to fields such as machine learning, bioinformatics, and finance. Through real-world case studies, hands-on examples, and guidance on implementing PCA using modern software tools and libraries, the book presents essential principles in quantum information theory and quantum algorithms, establishing the groundwork necessary to comprehend how quantum computing may expedite and improve PCA procedures. This work examines quantum algorithms for matrix decomposition, analyzes the computational benefits of quantum PCA compared to classical approaches, and showcases real applications in quantum machine learning, encryption, and quantum chemistry. Ultimately, this book will serve as a valuable resource for researchers, students, and professionals looking to the future of high-dimensional data analysis and how to apply efficient, scalable methods to PCA in their work.
Principal component analysis (PCA) is a widely used statistical tool for extracting low-dimensional structures underlying multivariate data. However, its application to high-dimensional data is limited due to its large computational time. While the conventional PCA algorithm requires polynomial time, using a quantum-inspired algorithm as a subroutine, we have implemented an algorithm that approximates it with computational time proportional to the logarithm of the input dimensionality. The computational efficiency and performance of the implemented algorithm, quantum-inspired PCA, are experimentally evaluated on synthetic and real datasets.
The paper reports a method for compressed representation of matrix data on the principles of quantum theory. The method is formalized as complex-valued matrix factorization based on standard singular value decomposition. The developed approach establishes a bridge between standard methods of semantic data analysis and quantum models of cognition and decision. According to the quantum theory, real-valued observable quantities are generated by wavefunctions being complex-valued vectors in multidimensional Hilbert-space. Wavefunctions are defined as superpositions of basis vectors encoding composition of semantic factors. Basis vectors are found by singular value decomposition of the initial data matrix transformed to a real-valued amplitude form. Phase-dependent superposition amplitudes are found to optimize approximation of the source data. The resulting model represents the observed real-valued data as generated from a small number of basis wavefunctions superposed with complex-valued coefficients. The method is tested for random matrices of sizes from 3 × 3 to 12 × 12 and dimensionality of latent Hilbert-space from 2 to 4. The best approximation is achieved by encoding latent factors in normalized complex-valued amplitude vectors interpreted as wavefunctions generating the data. In terms of approximation fitness, the developed method surpasses standard truncated SVD of the same dimensionality. The mean advantage over the considered range of parameters is 22 %. The method permits cognitive interpretation in accord with the existing quantum models of cognition and decision. The method can be integrated in the algorithms of semantic data analysis including natural language processing. In these tasks, the obtained improvement of approximation translates to the increased precision of similarity measures, principal component analysis, advantage in classification, and document ranking methods. Integration with quantum models of cognition and decision is expected to boost methods of artificial intelligence and machine learning improving imitation of natural thinking.