Quantum mean centering for block-encoding-based quantum algorithm

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