Obtaining A Linear Combination of the Principal Components of a Matrix on Quantum Computers
This work addresses the challenge of efficient data analysis for quantum computing applications, representing an incremental advancement in quantum algorithms.
The paper tackles the problem of performing principal component analysis on quantum computers by developing a method to obtain eigenvectors associated with the largest eigenvalues, enabling dimension reduction or feature extraction in quantum settings.
Principal component analysis is a multivariate statistical method frequently used in science and engineering to reduce the dimension of a problem or extract the most significant features from a dataset. In this paper, using a similar notion to the quantum counting, we show how to apply the amplitude amplification together with the phase estimation algorithm to an operator in order to procure the eigenvectors of the operator associated to the eigenvalues defined in the range $\left[a, b\right]$, where $a$ and $b$ are real and $0 \leq a \leq b \leq 1$. This makes possible to obtain a combination of the eigenvectors associated to the largest eigenvalues and so can be used to do principal component analysis on quantum computers.