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.

翻译：量子过程层析，即估计未知量子信道的任务，是量子信息理论中的一个核心问题。一个长期悬而未决的开放问题是：确定在钻石范数（区分量子过程的标准度量）下学习未知信道所需的最优信道使用次数。尽管量子态层析的类似问题在过去几十年中已在纯态和混合态两种场景下得到解决，但对于一般的量子信道，除了幺正信道这一特殊情况外，该问题在很大程度上仍然悬而未决。本文设计了一种算法，表明任何输入和输出维度分别为 $d_{\mathrm{in}},d_{\mathrm{out}}$ 且 Kraus 秩至多为 $k$ 的信道，都可以使用 $Θ(d_{\mathrm{in}}d_{\mathrm{out}}k)$ 次信道使用，以钻石范数学习到恒定精度；我们通过一个匹配的下界证明该标度是最优的。更一般地，达到 $\varepsilon$ 精度需要 $O(d_{\mathrm{in}}d_{\mathrm{out}}k/\varepsilon^{2})$ 次信道使用。由于量子信道将量子态、幺正算子和等距算子作为特例包含在内，我们的协议为相应的层析任务提供了一个统一的框架；特别是，它首次为等距算子和二元测量层析提供了最优协议，并且恢复了固定秩量子态在迹距离下的最优层析方案。我们的方法将信道层析简化为纯态层析：我们使用信道来制备其 Choi 态的副本，并行地纯化它们，并对得到的纯化态运行样本最优的纯态层析；然后我们证明，由此诱导的钻石范数误差在本质上与估计（纯化后的）Choi 态的迹距离误差成线性比例关系。我们还通过证明自适应性不会改善量子信道层析的维度最优查询复杂度，解决了一个开放性问题。