Recurrence Plot (RP) is a method employed to analyze the periodicity, chaoticity, and nonlinear characteristics of complex systems. Quantum Principal Component Analysis (QPCA), on the other hand, achieves dimensionality reduction of sample data using density matrices based on quantum circuits. We improve the distance threshold function of the recurrence plot algorithm using a density operator conceptually equivalent to the covariance matrix, integrate it with quantum circuits, and thereby develop a Quantum Recurrence Plot (QRP) algorithm. This algorithm achieves ultra-high efficiency in parallel computing, reduces computational costs, and simultaneously upgrades the traditional grayscale recurrence plot to colored heatmaps, enabling a better revelation of the system's dynamical characteristics.
Recurrence Plot (RP) is a method employed to analyze the periodicity, chaoticity, and nonlinear characteristics of complex systems. Quantum Principal Component Analysis (QPCA), on the other hand, achieves dimensionality reduction of sample data using density matrices based on quantum circuits. We improve the distance threshold function of the recurrence plot algorithm using a density operator conceptually equivalent to the covariance matrix, integrate it with quantum circuits, and thereby develop a Quantum Recurrence Plot (QRP) algorithm. This algorithm achieves ultra-high efficiency in parallel computing, reduces computational costs, and simultaneously upgrades the traditional grayscale recurrence plot to colored heatmaps, enabling a better revelation of the system's dynamical characteristics.
In this article, we propose a low-complexity quantum principal component analysis (qPCA) algorithm. Similar to the state-of-the-art qPCA, it achieves dimension reduction by extracting principal components of the data matrix, rather than all components of the data matrix, to quantum registers, so that the samples of measurement required can be reduced considerably. Both our qPCA and Lin’s qPCA are based on quantum singular-value thresholding (QSVT). The key of Lin’s qPCA is to combine QSVT, and modified QSVT is to obtain the superposition of the principal components. The key of our algorithm, however, is to modify QSVT by replacing the rotation-controlled operation of QSVT with the controlled-notoperation to obtain the superposition of the principal components. As a result, this small trick makes the circuit much simpler. Particularly, the proposed qPCA requires three phase estimations, while the state-of-the-art qPCA requires five phase estimations. Since the runtime of qPCA mainly comes from phase estimations, the proposed qPCA achieves a runtime of roughly 3/5 of that of the state of the art. We simulate the proposed qPCA on the IBM quantum computing platform, and the simulation result verifies that the proposed qPCA yields the expected quantum state.
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.
Quantum principal component analysis (qPCA) is the quantum version of principal component analysis (PCA). In this paper, based on the quantum singular value threshold (qSVT), we propose an exact quantum principal component analysis algorithm, which screens the data components through the threshold, rather than output all components of data. Compared with other improved qPCA algorithms, our proposed algorithm does not require to adjust the parameters to obtain estimated results. Instead, it yields exact results directly, and the quantum circuit designed is simpler because almost half of the quantum gates are reduced. We implemented our qPCA algorithm on the IBM quantum computing platform: IBM Quantum Experience, and the experimental results verified correctness of our algorithm.
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 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 commonly formulated as the extraction of eigenvalues and eigenvectors of a covariance-encoded density operator. Yet in many qPCA settings the practical goal is simpler: projection onto the dominant spectral subspace. Here we introduce a projection-first framework, the Filtered Spectral Projection Algorithm (FSPA), which bypasses explicit eigenvalue estimation while preserving the relevant spectral structure. FSPA amplifies any nonzero warm-start overlap with the leading subspace and remains robust in small-gap and near-degenerate regimes, without artificial symmetry breaking in the absence of bias. We show that FSPA achieves an oracle complexity O((log(1/ε)+log(1/∣a1∣2))/log(λ1/λ2)),which is tight by a matching lower bound, establishing it as an\emph{optimal} projection primitive. We derive a convergence rate for degenerate spectra, give a circuit resource analysis with n+O(1) qubit overhead independent of system dimension, and extend the method to threshold spectral projection, Threshold-FSPA, which converges in O(log(1/ε)) calls when the threshold lies between eigenvalues. In the density matrix exponentiation access model, FSPA gives an exponential copy-complexity advantage over classical methods. For classical datasets, we show that for amplitude-encoded centered data the ensemble density matrix ρ=∑ipi∣ψi⟩⟨ψi∣ equals the covariance matrix. Numerical tests on chemistry density matrices, noisy circuit outputs, Breast Cancer Wisconsin, handwritten Digits, and 1--4-qubit scalability confirm the theory. A minimal Qiskit implementation validates magnitude invariance, signal amplification, and no spurious symmetry breaking. These results establish FSPA as an optimal and deployable quantum spectral projection primitive.