Quantum algorithms for nearest-neighbor methods for supervised and unsupervised learning

We present quantum algorithms for performing nearest-neighbor learning and k--means clustering. At the core of our algorithms are fast and coherent quantum methods for computing the Euclidean distance both directly and via the inner product which we couple with methods for performing amplitude estimation that do not require measurement. We prove upper bounds on the number of queries to the input data required to compute such distances and find the nearest vector to a given test example. In the worst case, our quantum algorithms lead to polynomial reductions in query complexity relative to Monte Carlo algorithms. We also study the performance of our quantum nearest-neighbor algorithms on several real-world binary classification tasks and find that the classification accuracy is competitive with classical methods.