Quantum Laplacian Eigenmap

Laplacian eigenmap algorithm is a typical nonlinear model for dimensionality reduction in classical machine learning. We propose an efficient quantum Laplacian eigenmap algorithm to exponentially speed up the original counterparts. In our work, we demonstrate that the Hermitian chain product proposed in quantum linear discriminant analysis (arXiv:1510.00113,2015) can be applied to implement quantum Laplacian eigenmap algorithm. Compared with classical Laplacian eigenmap algorithm which requires polynomial time to solve dimensionality reduction, our algorithm is able to provide an exponential speedup.