  Within the framework of statistical learning theory it is possible to bound
the minimum number of samples required by a learner to reach a target accuracy.
We show that if the bound on the accuracy is taken into account, quantum
machine learning algorithms for supervised learning---for which statistical
guarantees are available---cannot achieve polylogarithmic runtimes in the input
dimension. We conclude that, when no further assumptions on the problem are
made, quantum machine learning algorithms for supervised learning can have at
most polynomial speedups over efficient classical algorithms, even in cases
where quantum access to the data is naturally available.
