Optimal learning of quantum channels in diamond distance
This solves a long-standing open problem in quantum information theory by providing tight bounds for channel tomography, a fundamental task that subsumes state, unitary, isometry, and measurement tomography.
The authors establish the optimal query complexity for learning an unknown quantum channel in diamond distance, showing that O(d_in d_out k / ε^2) channel uses suffice and Ω(d_in d_out k) are necessary at constant accuracy, with a separate Ω(1/ε^2) for non-minimal Kraus rank. Their algorithm reduces channel tomography to pure-state tomography via Choi state preparation and achieves optimal scaling.
Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory. A long-standing open question is how many uses of an unknown channel are required to learn it in diamond distance, the standard metric for distinguishing quantum processes. While quantum state tomography is well understood, for general channels the problem remained open beyond the unitary case. Here we establish the query complexity of channel tomography with optimal dependence on the dimension parameters, at any fixed constant accuracy. We design an algorithm showing that any channel with input/output dimensions $d_{\mathrm{in}},d_{\mathrm{out}}$ and Kraus rank at most $k$ can be learned to accuracy $\varepsilon$ using $O(d_{\mathrm{in}}d_{\mathrm{out}}k/\varepsilon^{2})$ channel uses. Conversely, we prove that $Ω(d_{\mathrm{in}}d_{\mathrm{out}}k)$ uses are necessary at constant accuracy and that, for non-minimal Kraus rank, a separate $Ω(1/\varepsilon^{2})$ contribution is unavoidable. Since channels subsume states, unitaries, isometries, and measurements as special cases, our protocol provides a unified framework for these tomography tasks, yielding new guarantees for isometry and measurement tomography while recovering known optimal scalings for state and unitary tomography. Our algorithm follows the natural strategy of performing optimal tomography on the Choi state. The main technical contribution is to show that this suffices to control the induced diamond-distance error, avoiding the dimension loss incurred by a naive conversion from Choi-state trace distance to channel diamond distance. The protocol uses the channel non-adaptively to prepare Choi-state copies, purifies them in parallel, and performs optimal pure-state tomography on the resulting purifications. Hence, we reduce channel tomography to pure-state tomography.