Quantum diffusion map for nonlinear dimensionality reduction
This work addresses the computational bottleneck in dimensionality reduction for large datasets, particularly in quantum physics applications, though it is incremental as it adapts an existing method to quantum computing.
The authors tackled the computational inefficiency of classical diffusion map algorithms for large datasets by proposing a quantum diffusion map (qDM) algorithm, achieving a total expected runtime proportional to N^2 polylog N with eigen-decomposition in O(log^3 N) time.
Inspired by random walk on graphs, diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a high-dimensional dataset. In recent years, among its many applications, DM has been successfully applied to discover relevant order parameters in many-body systems, enabling automatic classification of quantum phases of matter. However, classical DM algorithm is computationally prohibitive for a large dataset, and any reduction of the time complexity would be desirable. With a quantum computational speedup in mind, we propose a quantum algorithm for DM, termed quantum diffusion map (qDM). Our qDM takes as an input $N$ classical data vectors, performs an eigen-decomposition of the Markov transition matrix in time $O(\log^3 N)$, and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total expected runtime proportional to $N^2 \text{polylog}\, N$. Lastly, quantum subroutines in qDM for constructing a Markov transition matrix, and for analyzing its spectral properties can also be useful for other random walk-based algorithms.