In the problem of quantum channel certification, we have black box access to a quantum process and would like to decide if this process matches some predefined specification or is $\varepsilon$-far from this specification. The objective is to achieve this task while minimizing the number of times the black box is used. Here, we focus on optimal incoherent strategies for two relevant extreme cases of channel certification. The first one is when the predefined specification is a unitary channel, e.g., a gate in a quantum circuit. In this case, we show that testing whether the black box is described by a fixed unitary operator in dimension $d$ or $\varepsilon$-far from it in the trace norm requires $\Theta(d/\varepsilon^2)$ uses of the black box. The second setting we consider is when the predefined specification is a completely depolarizing channel with input dimension $d_{\text{in}}$ and output dimension $d_{\text{out}}$. In this case, we prove that, in the non-adaptive setting, $\tilde{\Theta}(d_{\text{in}}^2d_{\text{out}}^{1.5}/\varepsilon^2)$ uses of the channel are necessary and sufficient to verify whether it is equal to the depolarizing channel or $\varepsilon$-far from it in the diamond norm. Finally, we prove a lower bound of $\Omega(d_{\text{in}}^2d_{\text{out}}/\varepsilon^2)$ for this problem in the adaptive setting. Note that the special case $d_{\text{in}} = 1$ corresponds to the well-studied quantum state certification problem.

翻译：在量子信道认证问题中，我们可对量子过程进行黑箱访问，并需判断该过程是否与预定义规范相匹配，或与该规范在$\varepsilon$距离内。目标是在最小化黑箱使用次数的前提下完成此任务。本文聚焦于两种极端信道认证情形下的最优非相干策略。第一种情形是预定义规范为酉信道（例如量子电路中的门）。在此情形下，我们证明：判断黑箱是否由维度$d$中的固定酉算子描述，或与迹范数下$\varepsilon$距离内的酉算子描述，需要$\Theta(d/\varepsilon^2)$次黑箱使用。第二种情形考虑预定义规范为完全退极化信道，其输入维度为$d_{\text{in}}$，输出维度为$d_{\text{out}}$。我们证明：在非自适应设置下，需$\tilde{\Theta}(d_{\text{in}}^2d_{\text{out}}^{1.5}/\varepsilon^2)$次信道使用，才能验证该信道是否等于退极化信道，或与钻石范数下$\varepsilon$距离内的信道。最后，我们证明自适应设置下该问题的下界为$\Omega(d_{\text{in}}^2d_{\text{out}}/\varepsilon^2)$。值得注意的是，特例$d_{\text{in}} = 1$对应已充分研究的量子态认证问题。