The speed of quantum and classical learning for performing the k-th root of NOT

We consider quantum learning machines -- quantum computers that modify themselves in order to improve their performance in some way -- that are trained to perform certain classical task, i.e. to execute a function which takes classical bits as input and returns classical bits as output. This allows a fair comparison between learning efficiency of quantum and classical learning machine in terms of the number of iterations required for completion of learning. We find an explicit example of the task for which numerical simulations show that quantum learning is faster than its classical counterpart. The task is extraction of the k-th root of NOT (NOT = logical negation), with k=2^m and m \in N. The reason for this speed-up is that classical machine requires memory of size log k=m to accomplish the learning, while the memory of a single qubit is sufficient for the quantum machine for any k.