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 $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O\left(\mathrm{~d}^{2} / \varepsilon\right)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)$ [via standard process tomography] or $O\left(\mathrm{~d}^{2.5} / \varepsilon\right)$ [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 $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $\Omega\left(\mathrm{d}^{2} / \varepsilon\right)$ 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.