Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory and a key primitive for characterising noisy quantum devices. 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 measure of worst-case distinguishability between quantum processes. Here we show that a quantum channel acting on a $d$-dimensional system can be estimated to accuracy $\varepsilon$ in diamond distance using $O(d^4/\varepsilon^2)$ channel uses. This scaling is essentially optimal, as it matches lower bounds up to logarithmic factors. Our analysis extends to channels with input and output dimensions $d_{\mathrm{in}}$ and $d_{\mathrm{out}}$ and Kraus rank at most $k$, for which $O(d_{\mathrm{in}} d_{\mathrm{out}} k/\varepsilon^2)$ channel uses suffice, interpolating between unitary and fully generic channels. As by-products, we obtain, to the best of our knowledge, the first essentially optimal strategies for operator-norm learning of binary POVMs and isometries, and we recover optimal trace-distance tomography for fixed-rank states. Our approach consists of using the channel only non-adaptively to prepare copies of the Choi state, purify them in parallel, perform sample-optimal pure-state tomography on the purifications, and analyse the resulting estimator directly in diamond distance via its semidefinite-program characterisation. While the sample complexity of state tomography in trace distance is by now well understood, our results finally settle the corresponding problem for quantum channels in diamond distance.

翻译：量子过程层析，即估计未知量子信道的任务，是量子信息理论的核心问题，也是表征含噪声量子设备的关键原语。一个长期存在的开放问题是确定在钻石距离（量子过程间最坏情况可区分性的标准度量）下学习未知信道所需的最优使用次数。本文证明，作用于$d$维系统的量子信道可通过$O(d^4/\varepsilon^2)$次信道使用，在钻石距离下以精度$\varepsilon$进行估计。该标度本质最优，因其与对数因子内的下界匹配。我们的分析可推广至输入输出维度分别为$d_{\mathrm{in}}$和$d_{\mathrm{out}}$、Kraus秩至多为$k$的信道，此时$O(d_{\mathrm{in}} d_{\mathrm{out}} k/\varepsilon^2)$次信道使用即足够，覆盖了酉信道与完全一般信道之间的情形。作为副产品，我们首次（据我们所知）获得了二元POVM与等距映射的算子范数学习的本质最优策略，并恢复了固定秩态在迹距离下的最优层析方案。我们的方法包括：仅非自适应地使用信道制备Choi态的副本，并行纯化这些副本，对纯化态执行样本最优的纯态层析，并通过其半定规划表征直接分析所得估计量在钻石距离下的性能。尽管迹距离下态层析的样本复杂度现已得到充分理解，我们的结果最终解决了钻石距离下量子信道的对应问题。