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 $\epsilon$ requires $\hat{I}\text{c◯}\left(\frac{d_A{d}_{B}r}{\epsilon \log (d_{B}r/\epsilon )}\right)$ channel queries when $d_A=r{d}_B$, and $\hat{I}\text{c◯}\left(\frac{d_A{d}_{B}r}{{\epsilon }^2\log (d_{B}r/\epsilon )}\right)$ channel queries when $d_A\le r{d}_B/2$. These lower bounds improve upon the best previous $\hat{I}\text{c◯}(d_A{d}_{B}r)$ bound by introducing explicit, near-optimal $\epsilon$-dependence. Moreover, when $d_A\le r{d}_B/2$, the lower bound is optimal up to a logarithmic factor. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.

29 pages, 2 figures