The era of big data has made traditional machine learning algorithms struggle with high-dimensional, nonlinear datasets due to high computational resource consumption, long training times, and limited generalization. Quantum computing, with its exponential parallelism from qubit superposition and entanglement, offers new hope for machine learning. Local Sensitive Discriminant Analysis (LSDA) algorithm is a widely used dimensionality reduction technique in machine learning. However, it is time-consuming when handling large-scale high-dimensional data. This paper proposed a quantum version of LSDA algorithm, which transforms the generalized eigenvalue problem in LSDA into a Hermitian eigenvalue problem, solved using quantum phase estimation and block-encoding techniques. Two post-processing strategies are provided to obtain low-dimensional data for quantum or classical machine learning tasks. The quantum LSDA algorithm obtains polynomial speedups compared to the classical LSDA in terms of the number of data points and dimensionality, showing the potential of quantum technique to boost machine learning efficiency.
The era of big data has made traditional machine learning algorithms struggle with high-dimensional, nonlinear datasets due to high computational resource consumption, long training times, and limited generalization. Quantum computing, with its exponential parallelism from qubit superposition and entanglement, offers new hope for machine learning. Local Sensitive Discriminant Analysis (LSDA) algorithm is a widely used dimensionality reduction technique in machine learning. However, it is time-consuming when handling large-scale high-dimensional data. This paper proposed a quantum version of LSDA algorithm, which transforms the generalized eigenvalue problem in LSDA into a Hermitian eigenvalue problem, solved using quantum phase estimation and block-encoding techniques. Two post-processing strategies are provided to obtain low-dimensional data for quantum or classical machine learning tasks. The quantum LSDA algorithm obtains polynomial speedups compared to the classical LSDA in terms of the number of data points and dimensionality, showing the potential of quantum technique to boost machine learning efficiency.
Abstract Dimensionality reduction (DR) techniques play an extremely critical role in the data mining and pattern recognition field. However, most DR approaches involve large-scale matrix computations, which cause too high running complexity to implement in the big data scenario efficiently. The recent developments in quantum information processing provide a novel path to alleviate this problem, where a potential quantum acceleration can be obtained comparing with the classical counterpart. Nevertheless, existing proposals for quantum DR methods faced the common dilemma of the nonlinear generalization owing to the intrinsic linear limitation of quantum computation. In this paper, an architecture to simulate the arbitrary nonlinear kernels on a universal quantum computer is illustrated and further propose the quantum kernel principal component analysis (QKPCA) algorithm. The key idea is employing the truncated Taylor expansion to approximate the arbitrary nonlinear kernel within the fixed error and then constructing the corresponding Hamiltonian simulation for the quantum phase estimation algorithm. It is demonstrated theoretically that the QKPCA is qualified for the nonlinear DR task while the exponential speedup is also maintained. In addition, this research has the potential ability to develop other quantum DR approaches and existing linear quantum machine learning models.
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.
Abstract Principal component analysis (PCA) is an important dimensionality reduction method in machine learning and data analysis. Recently, the quantum version of PCA has been established to diagonalize quantum states. Although these quantum algorithms promise quantum advantages, they require substantial resources beyond the reach of state-of-the-art quantum technologies. This work aims to reduce resource requirements and improve the efficiency of quantum PCA. Assuming that the quantum state is accessed through a purified quantum query model and a sampling model, we propose quantum algorithms that use minimal resource requirements for ancillary qubits to reveal properties of eigenvectors and eigenvalues of a state. In particular, we show that estimating eigenvalue λ with error ε and success probability larger than λ ( 1 − η ) requests a query complexity O ~ ( ϵ − 1 ) and a sample complexity O ~ ( ϵ − 2 η − 1 ) , respectively. To our knowledge, our result is the first quantum speedup that achieves asymptotic linear scaling in 1 / ϵ for quantum PCA. As applications, we discuss estimating the minimum relative entropy of entanglement of bipartite pure-states and performing quantum state discrimination tasks. We show that quantum speedups are maintained when the pure state has a low Schmidt number and states of discrimination have a low rank. This study opens up a new quantum PCA method for high-dimensional quantum data analysis and discusses its application in quantum information processing tasks.
In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil (G,S). As a proof of principle, we implement our algorithm to solve ${2}^{5}\ifmmode\times\else\texttimes\fi{}{2}^{5}$ generalized eigenvalue problems. Finally, our results offer two optional outputs with quantum or classical form, which can be directly applied in another quantum or classical machine learning process.
Quantum computers acting as ML accelerators have attracted considerable interest as a promising avenue for addressing the limitations of classical computing confronted with solving complex, high-dimensional problems and handling large data sets. The most exciting aspect of quantum computing lies in its quantum advantage via techniques such as quantum parallelism, superposition, and entanglement which allows not only for rapid data processing, but also a means to resolve complex optimization problems beyond the capacity of classical algorithms. Nevertheless, the usage of quantum computing in machine learning is still a fledgling domain, with challenges concerning quantum noise, hardware restrictions, and algorithmic development. In particular, this research is concerned with the development of and study on hybrid quantum-classical algorithms that target to improve ML models in terms of computational capabilities by making use of the best of both worlds. To address this, researchers are developing hybrid algorithms that leverage the strengths of quantum computing for specific tasks, like optimization and pattern recognition, while still relying on classical computing for more flexible and larger-scale processing, ultimately achieving more efficient machine learning systems that can handle large datasets and complex problem-solving more effectively. It includes potential hybrid approaches like quantum-enhanced optimization algorithms, quantum neural networks, and quantum support vector machines, and examines how such algorithms can be used for practical ML problems including data classification, regression and clustering. Furthermore, the paper proposes that hybrid quantum algorithms offer several advantages over their classical counterparts, such as faster convergence rates, higher accuracy in high-dimensional spaces, and improved handling of non-linear and complex problem landscapes. Key advancements in the area of data processing speed and efficiency in problem-solving with and without quantum in the machine learning workflow are also highlighted in the research through comparative analysis and simulation studies. Lastly, the paper concludes by outlining the obstacles yet to be surmounted and the opportunities that lie ahead for hybrid quantum-classical algorithms, such as further improvement of quantum hardware and error mitigation methods, as well as the design of new algorithmic paradigms that seamlessly integrates the classical and quantum computing spaces.
Abstract - The exponential growth of large-scale data has created significant challenges in storage, processing, and optimization that classical computing systems struggle to address efficiently. Quantum Machine Learning (QML) has emerged as a promising paradigm that integrates quantum computing concepts with machine learning techniques to tackle complex big data optimization problems. By utilizing quantum features such as superposition, entanglement, and parallelism, QML algorithms can accelerate tasks like clustering, classification, and dimensionality reduction, which are traditionally computationally demanding. Recent advancements indicate strong potential for quantum-inspired optimization in fields including cybersecurity, financial portfolio management, Industry 4.0, and healthcare analytics. This paper provides a comprehensive survey of QML algorithms designed for big data optimization, examining their advantages, limitations, and application domains. Additionally, it discusses the performance trade-offs associated with hybrid quantum–classical frameworks and highlights future research directions aimed at creating secure, scalable, and resource-efficient quantum solutions for large-scale, data-driven decision-making. As data volumes continue to expand at extremely rapid rates, processing big data with traditional machine learning (ML) techniques has become increasingly challenging. ML models often struggle with issues related to computational power, efficient parameter tuning, accurate model selection, and maintaining high performance when dealing with massive datasets. Deep learning approaches—such as convolutional neural networks (CNNs)—demand substantial computational resources to train on large-scale supervised learning tasks. Moreover, the difficulty of training these networks grows significantly as the size and complexity of the datasets increase. In response to these limitations, Quantum Machine Learning (QML) has emerged as a promising research domain at the intersection of quantum computing and machine learning. Quantum computers operate fundamentally differently from classical systems, using the principles of quantum mechanics to encode and process information. As a result, quantum computational methods have the potential to address certain problems that are computationally intensive or infeasible for classical machines. Index Terms— Quantum Machine Learning (QML); Big Data Optimization; Quantum Algorithms; Hybrid Quantum-Classical Models; Quantum Computing; Data Analytics