  Nonparametric learning is able to make reliable predictions by extracting
information from similarities between a new set of input data and all samples.
Here we point out a quantum paradigm of nonparametric learning which offers an
exponential speedup over the sample size. By encoding data into quantum feature
space, similarity between the data is defined as an inner product of quantum
states. A quantum training state is introduced to superpose all data of
samples, encoding relevant information for learning in its bipartite
entanglement spectrum. We demonstrate that a trained state for prediction can
be obtained by entanglement spectrum transformation, using quantum matrix
toolbox. We further work out a feasible protocol to implement the quantum
nonparametric learning with trapped ions, and demonstrate the power of quantum
superposition for machine learning.
