Covariance Matrix Preparation for Quantum Principal Component Analysis
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 ma...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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American Physical Society
2022-09-01
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| Series: | PRX Quantum |
| Online Access: | http://doi.org/10.1103/PRXQuantum.3.030334 |
| _version_ | 1797998726909788160 |
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| author | Max Hunter Gordon M. Cerezo Lukasz Cincio Patrick J. Coles |
| author_facet | Max Hunter Gordon M. Cerezo Lukasz Cincio Patrick J. Coles |
| author_sort | Max Hunter Gordon |
| collection | DOAJ |
| description | 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 {p_{i},|ψ_{i}⟩}, then one can easily prepare the ensemble average density matrix ρ[over ¯]=[under ∑]ip_{i}|ψ_{i}⟩⟨ψ_{i}|. We first show that ρ[over ¯] 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[over ¯], and hence ρ[over ¯] 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. |
| first_indexed | 2024-04-11T10:53:19Z |
| format | Article |
| id | doaj.art-7a92e1fe01a64f758df7eaf52ab15e2f |
| institution | Directory Open Access Journal |
| issn | 2691-3399 |
| language | English |
| last_indexed | 2024-04-11T10:53:19Z |
| publishDate | 2022-09-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | PRX Quantum |
| spelling | doaj.art-7a92e1fe01a64f758df7eaf52ab15e2f2022-12-22T04:28:51ZengAmerican Physical SocietyPRX Quantum2691-33992022-09-013303033410.1103/PRXQuantum.3.030334Covariance Matrix Preparation for Quantum Principal Component AnalysisMax Hunter GordonM. CerezoLukasz CincioPatrick J. ColesPrincipal 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 {p_{i},|ψ_{i}⟩}, then one can easily prepare the ensemble average density matrix ρ[over ¯]=[under ∑]ip_{i}|ψ_{i}⟩⟨ψ_{i}|. We first show that ρ[over ¯] 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[over ¯], and hence ρ[over ¯] 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.http://doi.org/10.1103/PRXQuantum.3.030334 |
| spellingShingle | Max Hunter Gordon M. Cerezo Lukasz Cincio Patrick J. Coles Covariance Matrix Preparation for Quantum Principal Component Analysis PRX Quantum |
| title | Covariance Matrix Preparation for Quantum Principal Component Analysis |
| title_full | Covariance Matrix Preparation for Quantum Principal Component Analysis |
| title_fullStr | Covariance Matrix Preparation for Quantum Principal Component Analysis |
| title_full_unstemmed | Covariance Matrix Preparation for Quantum Principal Component Analysis |
| title_short | Covariance Matrix Preparation for Quantum Principal Component Analysis |
| title_sort | covariance matrix preparation for quantum principal component analysis |
| url | http://doi.org/10.1103/PRXQuantum.3.030334 |
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