Abstract Quantum principal component analysis is a dimensionality reduction method to select the significant features of a dataset. A classical method finds the solution in polynomial time, but when the dimension of feature space scales exponentially, it is inefficient to compute the matrix exponentiation of the covariance matrix. The quantum method uses density matrix exponentiation to find principal components with exponential speedup. We enhance the existing algorithm that applies amplitude amplification using range-based static selection of eigenstates on the output of phase estimation. So, we propose an equivalent quantum method with the same complexity using a dynamic selection of eigenstates. Our algorithm can efficiently find phases of equally likely eigenvalues based on the similarity scores. It obtains principal components associated with highly probable larger eigenvalues. We analyze these methods on various factors to justify the resulting complexity of a proposed method as effective in quantum counterparts.
Abstract Quantum principal component analysis is a dimensionality reduction method to select the significant features of a dataset. A classical method finds the solution in polynomial time, but when the dimension of feature space scales exponentially, it is inefficient to compute the matrix exponentiation of the covariance matrix. The quantum method uses density matrix exponentiation to find principal components with exponential speedup. We enhance the existing algorithm that applies amplitude amplification using range-based static selection of eigenstates on the output of phase estimation. So, we propose an equivalent quantum method with the same complexity using a dynamic selection of eigenstates. Our algorithm can efficiently find phases of equally likely eigenvalues based on the similarity scores. It obtains principal components associated with highly probable larger eigenvalues. We analyze these methods on various factors to justify the resulting complexity of a proposed method as effective in quantum counterparts.
Quantum principal component analysis (QPCA) is analogous to the classical method with additive speedup. This is based on the unsupervised dimensionality reduction mechanism that transforms exponentially high dimensional data points into low dimensional space by retaining significant features with the largest variance. To our knowledge, quantum computations are highly exponential and supports implicit parallelism. Therefore, quantum method for finding principal components are proved to achieve exponential speedup over existing classical solution. An advantage of quantum superposition is utilized for the Eigen decomposition of density matrix which is identical to covariance, however, a way through which its computation happen, is truly an interesting factor. This paper presents basic methodological framework to design QPCA algorithm. Our framework includes the methods for quantum data encoding, quantum memory processing, and the requisite quantum circuit implementations to realize QPCA algorithms. In addition, we categorized existing QPCA methods in view of recent advances. We then proposed the possible advancements that can benefit to design quantum based efficient algorithms for principal component analysis. At the last, suggested concepts are compared with existing methods based on complexities and other factors to justify our proposal as significant in perspective to future directions.
Abstract Performing the eigendecomposition of the covariance matrix of the dataset is of great significance in the field of machine learning. However, classical operations will become time-consuming when involving large data sets. In this paper, in order to address this problem, we design an efficient quantum algorithm to prepare the covariance matrix state by means of quantum amplitude estimation. After that, we research on its application in principal component analysis and Mahalanobis distance calculation. Specifically, we obtain the transformation matrix for quantum principal component analysis based on the singular value estimation algorithm and the amplitude amplification algorithm. Furthermore, we invoke the quantum matrix inversion algorithm to calculate the Mahalanobis distance. The final complexity analysis shows that our proposed algorithms can achieve speedup compared to their classical counterparts under certain conditions.
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.
Principal component analysis (PCA) is a dimensionality reduction method in data analysis that involves diagonalizing the covariance matrix of the dataset. Recently, quantum algorithms have been formulated for PCA based on diagonalizing a density matrix. These algorithms assume that the covariance matrix can be encoded in a density matrix, but a concrete protocol for this encoding has been lacking. Our work aims to address this gap. Assuming amplitude encoding of the data, with the data given by the ensemble {pi,∣ψi⟩}, then one can easily prepare the ensemble average density matrix ρ=∑ipi∣ψi⟩⟨ψi∣. We first show that ρ is precisely the covariance matrix whenever the dataset is centered. For quantum datasets, we exploit global phase symmetry to argue that there always exists a centered dataset consistent with ρ, and hence ρ can always be interpreted as a covariance matrix. This provides a simple means for preparing the covariance matrix for arbitrary quantum datasets or centered classical datasets. For uncentered classical datasets, our method is so-called "PCA without centering", which we interpret as PCA on a symmetrized dataset. We argue that this closely corresponds to standard PCA, and we derive equations and inequalities that bound the deviation of the spectrum obtained with our method from that of standard PCA. We numerically illustrate our method for the MNIST handwritten digit dataset. We also argue that PCA on quantum datasets is natural and meaningful, and we numerically implement our method for molecular ground-state datasets.
Principal component analysis (PCA) has been widely adopted to reduce the dimension of data while preserving the information. The quantum version of PCA (qPCA) can be used to analyze an unknown low-rank density matrix by rapidly revealing the principal components of it, i.e., the eigenvectors of the density matrix with the largest eigenvalues. However, because of the substantial resource requirement, its experimental implementation remains challenging. Here, we develop a resonant analysis algorithm with minimal resource for ancillary qubits, in which only one frequency-scanning probe qubit is required to extract the principal components. In the experiment, we demonstrate the distillation of the first principal component of a 4 × 4 density matrix, with an efficiency of 86.0% and a fidelity of 0.90. This work shows the speedup ability of quantum algorithm in dimension reduction of data and thus could be used as part of quantum artificial intelligence algorithms in the future.
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.