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Covariance Matrix Preparation for Quantum Principal Component Analysis

Max Hunter Gordon, Marco Cerezo, Łukasz Cincio, Patrick J. Coles

Published Sep 7, 2022• 28 citations
Publisher

Abstract

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$\rangle$}, then one can easily prepare the ensemble average density matrix $\overline{ρ}$ =Σipi|ψi$\rangle$$\langle$ψi|. We first show that $\overline{ρ}$ 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 $\overline{ρ}$ , and hence $\overline{ρ}$ 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 Modified National Institute of Standards and Technology (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.

DOI
10.1103/prxquantum.3.030334

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International Conference on Computing, Communication and Networking Technologies (ICCCNT)•Jul 6, 2021•3 citations
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⟩\rangle⟩}, then one can easily prepare the ensemble average density matrix ρ‾\overline{ρ}ρ​ =Σipi|ψi\rangle$$\langleψi|. We first show that ρ‾\overline{ρ}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‾\overline{ρ}ρ​ , and hence ρ‾\overline{ρ}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 Modified National Institute of Standards and Technology (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) 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⟩}\{p_i,| ψ_i \rangle\}{pi​,∣ψi​⟩}, then one can easily prepare the ensemble average density matrix ρ‾=∑ipi∣ψi⟩⟨ψi∣\overlineρ = \sum_i p_i |ψ_i\rangle \langle ψ_i |ρ​=∑i​pi​∣ψi​⟩⟨ψi​∣. We first show that ρ‾\overlineρρ​ 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‾\overlineρρ​, and hence ρ‾\overlineρρ​ 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.
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 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))\mathcal{O}((\log(1/ε)+\log(1/|a_1|^2))/\log(λ_1/λ_2))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)n+\mathcal{O}(1)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))\mathcal{O}(\log(1/ε))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∣ρ=\sum_i p_i|ψ_i\rangle\langleψ_i|ρ=∑i​pi​∣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.
Principal component analysis (PCA) is traditionally implemented through a covariance or kernel matrix, leading-eigenvector extraction, and hard rank-kkk projection. These steps can be computationally costly in high-dimensional and quantum-data settings, sensitive to small eigengaps, and unnecessary when downstream tasks only require principal-subspace scores. Such score-based objectives are important in applications such as anomaly detection, spectral-energy profiling, and other postselection tasks. To address these needs, we introduce a measurement-based soft PCA framework replacing the hard top-kkk projector with an entropy-regularized Fermi--Dirac filter. This filter is the unique optimizer of an entropy-regularized variational formulation of PCA and converges to the classical PCA projector in the zero-temperature limit. This filter has a direct interpretation as a quantum measurement, which naturally suggests a quantum approach. For centered covariance operators represented by quantum feature states, a single fixed circuit, together with threshold calibration, accesses all optimal filters for different rank budgets or retained-variance levels without rank-dependent circuit updates or eigenvector recovery. For new inputs, the same calibrated quantum circuit yields soft principal subspace scores, spectral energy profiles, and postselected filtered states. The required centering of both training and test data is performed coherently inside the quantum protocol, which is particularly important for quantum data where no classical feature vectors or centered Gram matrix are directly available. By reframing PCA as a calibrated measurement task, this framework bypasses the need for iterative eigenvector extraction and achieves a dimension-independent sample complexity O(η−2)O(η^{-2})O(η−2) for normalized fractional-rank or retained variance scoring at additive accuracy ηηη.
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.