We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a $\textsf{d}$-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O(\textsf{d}^2/\varepsilon)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O(\textsf{d}^3/\varepsilon^2)$ [via standard process tomography] or $O(\textsf{d}^{2.5}/\varepsilon)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to "bootstrap" an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $\Omega(\textsf{d}^2/\varepsilon)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.

翻译：我们考虑酉量子信道的过程层析成像。在可访问作用于$\textsf{d}$维量子系统的未知酉信道的前提下，我们的目标是输出一个经典描述的酉算子，其在钻石范数下与未知酉算子的误差不超过$\varepsilon$。我们设计了一种算法，通过$O(\textsf{d}^2/\varepsilon)$次未知信道调用（仅需一个量子系统）即可实现误差$\varepsilon$。该结果优于先前方法——标准过程层析成像需$O(\textsf{d}^3/\varepsilon^2)$次调用，而杨、伦纳和基里贝拉（PRL 2020）的方法需$O(\textsf{d}^{2.5}/\varepsilon)$次调用。为证明该结果，我们引入了一种简单技术，可将仅能产生常数误差估计的算法"引导"为具有海森堡标度的$\varepsilon$误差估计。最后，我们证明了互补的下界：即便可访问未知酉算子的逆或受控版本，估计仍需$\Omega(\textsf{d}^2/\varepsilon)$次调用。这表明我们的算法同时具有最优查询复杂度和最优空间复杂度。