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arXiv · 2208.02143

Quantum mean centering for block-encoding-based quantum algorithm

Hai-Ling Liu·Chao-Hua Yu·Lin-Chun Wan·Su-Juan Qin·Fei Gao·Qiao-Yan Wen
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Abstract

Mean Centering (MC) is an important data preprocessing technique, which has a wide range of applications in data mining, machine learning, and multivariate statistical analysis. When the data set is large, this process will be time-consuming. In this paper, we propose an efficient quantum MC algorithm based on the block-encoding technique, which enables the existing quantum algorithms can get rid of the assumption that the original data set has been classically mean-centered. Specifically, we first adopt the strategy that MC can be achieved by multiplying by the centering matrix $C$, i.e., removing the row means, column means and row-column means of the original data matrix $X$ can be expressed as $XC$, $CX$ and $CXC$, respectively. This allows many classical problems involving MC, such as Principal Component Analysis (PCA), to directly solve the matrix algebra problems related to $XC$, $CX$ or $CXC$. Next, we can employ the block-encoding technique to realize MC. To achieve it, we first show how to construct the block-encoding of the centering matrix $C$, and then further obtain the block-encodings of $XC$, $CX$ and $CXC$. Finally, we describe one by one how to apply our MC algorithm to PCA and other algorithms.

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Hai-Ling Liu, Chao-Hua Yu, Lin-Chun Wan, Su-Juan Qin, Fei Gao, Qiao-Yan Wen. 2022-08-03. Quantum mean centering for block-encoding-based quantum algorithm. https://doi.org/10.1016/j.physa.2022.128227

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