Query-optimal estimation of unitary channels in diamond distance
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We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a -dimensional qudit, we aim to output a classical description of a unitary that is ε-close to the unknown unitary in diamond norm. We design an algorithm achieving error ε using O(^2/ε) applications of the unknown channel and only one qudit. This improves over prior results, which use O(^3/ε^2) [via standard process tomography] or O(^2.5/ε) [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 ε-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires Ω(^2/ε) 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.
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