Improved Lower Bounds for Learning Quantum Channels in Diamond Distance

Abstract: We prove that learning an unknown quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ to diamond distance $\varepsilon$ requires $ Ω!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ queries. This improves the best previous $Ω(d_A d_B r)$ bound by introducing explicit $\varepsilon$-dependence, with a scaling in $\varepsilon$ that is near-optimal when $d_A=rd_B$ but not tight in general. The proof constructs an ensemble of channels that are well-separated in diamond norm yet admit Stinespring isometries that are close in operator norm.