Quantum spectral clustering

Abstract

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