  Machine-learning tasks frequently involve problems of manipulating and
classifying large numbers of vectors in high-dimensional spaces. Classical
algorithms for solving such problems typically take time polynomial in the
number of vectors and the dimension of the space. Quantum computers are good at
manipulating high-dimensional vectors in large tensor product spaces. This
paper provides supervised and unsupervised quantum machine learning algorithms
for cluster assignment and cluster finding. Quantum machine learning can take
time logarithmic in both the number of vectors and their dimension, an
exponential speed-up over classical algorithms.
