  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 $\{p_i,|
\psi_i \rangle\}$, then one can easily prepare the ensemble average density
matrix $\overline{\rho} = \sum_i p_i |\psi_i\rangle \langle \psi_i |$. We first
show that $\overline{\rho}$ 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
$\overline{\rho}$, and hence $\overline{\rho}$ 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 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.

Quantum Physics;Machine Learning;Machine Learning