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

TL;DR

This paper demonstrates that quantum learning machines can outperform classical ones in learning the k-th root of NOT task, requiring less memory and fewer iterations, as shown through explicit examples and simulations.

Contribution

It provides the first explicit example where quantum learning outperforms classical learning in terms of speed and memory efficiency for a specific logical task.

Findings

Quantum learning is faster than classical learning for the k-th root of NOT.

Quantum machines require less memory (single qubit) than classical machines (log k bits).

Simulations confirm the quantum speed-up in learning efficiency.

Abstract

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.