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Optimal learning of quantum channels in diamond distance

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 to determine the optimal number of uses of an unknown channel required to learn it in diamond distance, the standard metric for distinguishing quantum processes. While the analogous problem of quantum state tomography has been settled over the past decades in both the pure- and mixed-state settings, for general quantum channels it remained largely open beyond the unitary case. Here we design an algorithm showing that any channel with input and output dimensions $d_{\mathrm{in}},d_{\mathrm{out}}$ and Kraus rank at most $k$ can be learned to constant accuracy in diamond distance using $Θ(d_{\mathrm{in}}d_{\mathrm{out}}k)$ channel uses, and we prove that this scaling is optimal via a matching lower bound. More generally, achieving accuracy $\varepsilon$ is possible with $O(d_{\mathrm{in}}d_{\mathrm{out}}k/\varepsilon^{2})$ channel uses. Since quantum channels subsume states, unitaries, and isometries as special cases, our protocol provides a unified framework for the corresponding tomography tasks; in particular, it yields the first optimal protocols for isometries and for binary measurement tomography, and it recovers optimal trace-distance tomography for fixed-rank states. Our approach reduces channel tomography to pure-state tomography: we use the channel to prepare copies of its Choi state, purify them in parallel, and run sample-optimal pure-state tomography on the resulting purifications; we then show that the induced diamond-distance error scales essentially linearly with the trace-distance error in estimating the (purified) Choi state. We also resolve an open question by showing that adaptivity does not improve the dimension-optimal query complexity of quantum channel tomography.