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.

翻译：量子过程层析成像（即估计未知量子通道的任务）是量子信息理论中的核心问题。一个长期悬而未决的问题是：在区分量子过程的标准度量——金刚石距离下，需要多少次未知通道的使用才能学习该通道。尽管量子态层析成像已被充分理解，但除酉通道外，一般通道的这一问题仍未解决。本文在任意固定常数精度下，建立了通道层析成像的查询复杂度与维度参数的最优依赖关系。我们设计了一个算法，表明任何输入/输出维度为$d_{\mathrm{in}}, d_{\mathrm{out}}$且Kraus秩至多为$k$的通道，可通过$O(d_{\mathrm{in}}d_{\mathrm{out}}k/\varepsilon^{2})$次通道使用，以精度$\varepsilon$进行学习。相反，我们证明在常数精度下至少需要$\Omega(d_{\mathrm{in}}d_{\mathrm{out}}k)$次使用，且对于非最小Kraus秩的情况，一个独立的$\Omega(1/\varepsilon^{2})$贡献不可避免。由于通道将量子态、酉操作、等距变换和测量作为特例包含在内，我们的协议为这些层析成像任务提供了统一框架，为等距变换和测量层析成像带来了新保证，并恢复了量子态和酉通道层析成像中已知的最优缩放。该算法遵循了对Choi态执行最优层析成像的自然策略。主要技术贡献在于证明了这足以控制诱导的金刚石距离误差，避免了从Choi态迹距离到通道金刚石距离的朴素转换所导致的维度损失。该协议非自适应地使用通道制备Choi态副本，并行对其纯化，并对所得纯化态执行最优纯态层析成像。因此，我们将通道层析成像约化为纯态层析成像。