Quantum spectral clustering

Spectral clustering is a powerful unsupervised machine learning algorithm for data with non convex or nested structures. With roots in graph theory, it uses the spectral properties of Laplacian matrix to project low-dimensional space where more efficient. Despite its success tasks, suffers practice from fast-growing running time $O(n^3)$, $n$ number points dataset. In this work we propose an end-to-end quantum performing clustering, extending works learning. The composed two parts: first efficient creation state corresponding projected matrix, and second consists applying existing analogue $k$-means algorithm. Both steps depend polynomially on clusters, as well precision parameters arising procedures, polylogarithmically dimension input vectors. Our numerical simulations show asymptotic linear growth when all terms are taken into account, significantly better than classical cubic growth. This opens path other graph-based algorithms, provides routines computation access Incidence, Adjacency, matrices graph.