Query-optimal estimation of unitary channels in diamond distance

Abstract: We will present an algorithm to learn an unknown unitary channel acting on a d-dimensional qudit to diamond-norm error ε, using O(d²/ε) applications of the unknown channel and only one qudit. This algorithm uses the optimal number of qudits and number of queries, even if one has access to the inverse or controlled versions of the unknown unitary. This improves over prior work, which achieves entanglement infidelity δ using O(d²/√δ) applications in parallel, thereby requiring Ω(d²) qudits. Based on joint work with Jeongwan Haah, Robin Kothari, and Ryan O'Donnell.