论文标题
基于块编码的量子算法的量子平均值
Quantum mean centering for block-encoding-based quantum algorithm
论文作者
论文摘要
平均居中(MC)是一种重要的数据预处理技术,在数据挖掘,机器学习和多元统计分析中具有广泛的应用。当数据集很大时,此过程将很耗时。在本文中,我们基于块编码技术提出了一种有效的量子MC算法,该算法使现有的量子算法可以摆脱以下假设,即原始数据集已经典均值为中心。具体来说,我们首先采用了可以通过将MC乘以矩阵$ c $来实现的策略,即删除了行均值,列均值和行柱的原始数据矩阵$ x $的均值分别表示为$ xc $,$ cx $,$ cx $和$ cxc $。这允许许多涉及MC的经典问题,例如主组件分析(PCA),可以直接解决与$ XC $,$ CX $或$ CXC $相关的矩阵代数问题。接下来,我们可以采用块编码技术来实现MC。为了实现这一目标,我们首先展示了如何构建中心矩阵$ c $的块编码,然后进一步获得$ xc $,$ cx $和$ cxc $的块编码。最后,我们一一描述如何将MC算法应用于PCA和其他算法。
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.