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 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.