Distributional property testing in a quantum world textabstractA fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speedups. In particular, we give fast quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon  von Neumann entropy of distributions. The distributions can be either classical or quantum, however our quantum algorithms require coherent quantum access to a process preparing the samples. Our results build on the recent technique of quantum singular value transformation, combined with more standard tricks such as divideandconquer. The presented approach is a natural fit for distributional property testing both in the classical and the quantum case, demonstrating the first speedups for testing properties of density operators that can be accessed coherently rather than only via sampling; for classical distributions our algorithms significantly improve the precision dependence of some earlier results.

Supervised learning with a quantum classifier using multilevel systems We propose a quantum classifier, which can classify data under the supervised learning scheme using a quantum feature space. The input feature vectors are encoded in a single quNit (a Nlevel quantum system), as opposed to more commonly used entangled multiqubit systems. For training, we use the much used quantum variational algorithmxe2x80x94a hybrid quantumxe2x80x93classical algorithm, in which the forward part of the computation is performed on a quantum hardware, whereas the feedback part is carried out on a classical computer. We introduce xe2x80x9csingleshot trainingxe2x80x9d, where all input samples belonging to the same class are used to train the classifier simultaneously. This significantly speeds up the training procedure and provides an advantage over classical machine learning classifiers. We demonstrate successful classification of popular benchmark datasets with our quantum classifier and compare its performance with respect to classical machine learning classifiers. We also show that the number of training parameters in our classifier is significantly less than the classical classifiers.