Quantum diffusion map for nonlinear dimensionality reduction

Inspired by random walks on graphs, the diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification low-dimensional data structure hidden in high-dimensional set. In recent years, among its many applications, DM has been successfully applied to discover relevant order parameters many-body systems, enabling classification quantum phases matter. However, classical algorithm computationally prohibitive for large set, and any reduction time complexity would be desirable. With computational speedup mind, we propose DM, termed (qDM). Our qDM takes as an input $N$ vectors, performs eigendecomposition Markov transition matrix $O({log}^{3}N)$, classically constructs via readout (tomography) eigenvectors, giving total expected runtime proportional ${N}^{2}\text{polylog}\phantom{\rule{0.16em}{0ex}}N$. Finally, subroutines constructing analyzing spectral properties can also useful other random-walk-based algorithms.