The speed of quantum and classical learning for performing the kth 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 that takes classical bits as input and returns classical bits as output. This allows a fair comparison between learning efficiency of quantum and classical learning machines 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 kth root of NOT (NOT = logical negation), with k=2m and... The reason for this speed-up is that the 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.We acknowledge support from the EC Project QAP (no. 015848), the Austrian Science Foundation FWF within projects no. P19570-N16, SFB and CoQuS no. W1210-N16, the Ministerio de Ciencia e Innovación (Fellowship BES-2006-13234) and the Instituto Carlos I for the use of computational resources. The collaboration is a part of an ÖAD/MNiSW program