Quantum Discriminant Analysis for Dimensionality Reduction and Classification

Abstract: We present quantum algorithms to efficiently perform discriminant analysis for dimensionality reduction and classification over an exponentially large input data set. Compared with the best-known classical algorithms, the quantum algorithms show an exponential speedup in both the number of training vectors $M$ and the feature space dimension $N$. We generalize the previous quantum algorithm for solving systems of linear equations [Phys. Rev. Lett. 103, 150502 (2009)] to efficiently implement a Hermitian chain product of $k$ trace-normalized $N \times N$ Hermitian positive-semidefinite matrices with time complexity of $O(\log (N))$. Using this result, we perform linear as well as nonlinear Fisher discriminant analysis for dimensionality reduction over $M$ vectors, each in an $N$-dimensional feature space, in time $O(p \text{polylog} (MN)/\epsilon ^{3})$, where $\epsilon $ denotes the tolerance error, and $p$ is the number of principal projection directions desired. We also present a quantum discriminant analysis algorithm for data classification with time complexity $O(\log (MN)/\epsilon^{3})$.