q-means: A quantum algorithm for unsupervised machine learning

Quantum information is a promising new paradigm for fast computations that can provide substantial speedups for many algorithms we use today. Among them, quantum machine learning is one of the most exciting applications of quantum computers. In this paper, we introduce q-means, a new quantum algorithm for clustering. It is a quantum version of a robust k-means algorithm, with similar convergence and precision guarantees. We also design a method to pick the initial centroids equivalent to the classical k-means++ method. Our algorithm provides currently an exponential speedup in the number of points of the dataset, compared to the classical k-means algorithm. We also detail the running time of q-means when applied to well-clusterable datasets. We provide a detailed runtime analysis and numerical simulations for specific datasets. Along with the algorithm, the theorems and tools introduced in this paper can be reused for various applications in quantum machine learning.