Quantum discriminant analysis for dimensionality reduction and classification
Iris Cong, Luming Duan · New Journal of Physics · 2016
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 (2009 Phys. Rev. Lett. 103 150502 ) to efficiently implement a Hermitian chain product of k trace-normalized N × N Hermitian positive-semidefinite matrices with time complexity of . 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 , where 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 .