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，我们通过黑盒访问一个量子过程，需要判断该过程是否符合预定义的规范，或与规范相差$\varepsilon$（在迹范数意义下）。目标是尽可能减少黑盒使用次数来完成这一任务。本文聚焦于信道认证中两种极端情况的最优非相干策略。第一种情况是预定义规范为酉信道（例如量子电路中的量子门）。在此情形下，我们证明：在$d$维空间中，判断黑盒是否由固定酉算子描述（或与之在迹范数意义上相差$\varepsilon$）需要$\Theta(d/\varepsilon^2)$次黑盒使用。第二种情况是预定义规范为具有输入维度$d_{\text{in}}$和输出维度$d_{\text{out}}$的完全退极化信道。在此情形下，我们证明：在非自适应设置中，使用$\tilde{\Theta}(d_{\text{in}}^2d_{\text{out}}^{1.5}/\varepsilon^2)$次信道查询足以（且必要）验证该信道是否等于退极化信道（或与之在金刚石范数意义上相差$\varepsilon$）。最后，我们证明该问题在自适应设置中的下限为$\Omega(d_{\text{in}}^2d_{\text{out}}/\varepsilon^2)$。特别地，当$d_{\text{in}} = 1$时，此问题退化为已被充分研究的量子态认证问题。