Query-optimal estimation of unitary channels in diamond distance

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is epsilon-close to the unknown unitary in diamond norm. We design an algorithm achieving error epsilon using O(d(2)/epsilon) applications of the unknown channel and only one qudit. This improves over prior results, which use O(d(3)/epsilon(2)) [via standard process tomography] or O(d(2.5)/epsilon) [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to "bootstrap" an algorithm that can produce constant-error estimates to one that can produce epsilon-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires Omega(d(2)/epsilon) applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.