Quantum-enhanced learning of rotations about an unknown direction

We design machines that learn how to rotate a quantum bit about an initially unknown direction, encoded in the state of a spin-j particle. We show that a machine equipped with a quantum memory of O(log j) qubits can outperform all machines with purely classical memory, even if the size of their memory is arbitrarily large. The advantage is present for every finite j and persists as long as the quantum memory is accessed for no more than O(j) times. We establish these results by deriving the ultimate performance achievable with purely classical memories, thus providing a benchmark that can be used to experimentally demonstrate the implementation of quantum-enhanced learning.