  Modern quantum machine learning (QML) methods involve variationally
optimizing a parameterized quantum circuit on a training data set, and
subsequently making predictions on a testing data set (i.e., generalizing). In
this work, we provide a comprehensive study of generalization performance in
QML after training on a limited number $N$ of training data points. We show
that the generalization error of a quantum machine learning model with $T$
trainable gates scales at worst as $\sqrt{T/N}$. When only $K \ll T$ gates have
undergone substantial change in the optimization process, we prove that the
generalization error improves to $\sqrt{K / N}$. Our results imply that the
compiling of unitaries into a polynomial number of native gates, a crucial
application for the quantum computing industry that typically uses
exponential-size training data, can be sped up significantly. We also show that
classification of quantum states across a phase transition with a quantum
convolutional neural network requires only a very small training data set.
Other potential applications include learning quantum error correcting codes or
quantum dynamical simulation. Our work injects new hope into the field of QML,
as good generalization is guaranteed from few training data.
