We study the complexity of testing properties of quantum channels. First, we show that testing identity to any channel $\mathcal N: \mathbb C^{d_{\mathrm{in}} \times d_{\mathrm{in}}} \to \mathbb C^{d_{\mathrm{out}} \times d_{\mathrm{out}}}$ in diamond norm distance requires $\Omega(\sqrt{d_{\mathrm{in}}} / \varepsilon)$ queries, even in the strongest algorithmic model that admits ancillae, coherence, and adaptivity. This is due to the worst-case nature of the distance induced by the diamond norm. Motivated by this limitation and other theoretical and practical applications, we introduce an average-case analogue of the diamond norm, which we call the average-case imitation diamond (ACID) norm. In the weakest algorithmic model without ancillae, coherence, or adaptivity, we prove that testing identity to certain types of channels in ACID distance can be done with complexity independent of the dimensions of the channel, while for other types of channels the complexity depends on both the input and output dimensions. Building on previous work, we also show that identity to any fixed channel can be tested with $\tilde O(d_{\mathrm{in}} d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ queries in ACID distance and $\tilde O(d_{\mathrm{in}}^2 d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ queries in diamond distance in this model. Finally, we prove tight bounds on the complexity of channel tomography in ACID distance.

翻译：我们研究了量子信道性质测试的复杂度。首先，我们证明在菱形范数距离下测试任意信道 $\mathcal N: \mathbb C^{d_{\mathrm{in}} \times d_{\mathrm{in}}} \to \mathbb C^{d_{\mathrm{out}} \times d_{\mathrm{out}}}$ 的恒等性需要 $\Omega(\sqrt{d_{\mathrm{in}}} / \varepsilon)$ 次查询，即使在允许辅助系统、相干性和自适应性的最强算法模型中也是如此。这是由于菱形范数所诱导的距离具有最坏情况特性。受此局限性和其他理论及实际应用的启发，我们引入了菱形范数的平均情况类比，称之为平均情况模拟菱形（ACID）范数。在最弱的无辅助系统、无相干性且无自适应性的算法模型中，我们证明了在ACID距离下测试某些类型信道的恒等性可以实现与信道维度无关的复杂度，而对于其他类型信道，复杂度则同时依赖于输入和输出维度。基于先前工作，我们还证明了在该模型中，任意固定信道的恒等性测试在ACID距离下需要 $\tilde O(d_{\mathrm{in}} d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ 次查询，在菱形距离下需要 $\tilde O(d_{\mathrm{in}}^2 d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ 次查询。最后，我们证明了在ACID距离下进行信道层析的复杂度紧界。