Improved Lower Bounds for Learning Quantum Channels in Diamond Distance

• URL: http://arxiv.org/abs/2601.04180v3
• Date: Tue, 13 Jan 2026 18:58:46 GMT
• Title: Improved Lower Bounds for Learning Quantum Channels in Diamond Distance
• Authors: Aadil Oufkir, Filippo Girardi,
• Abstract summary: 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 $!left( fracd_A d_B rvarepsilon log(d_B r / varepsilon) right)$ channel queries when $d_A= rd_B$, and $!left( fracd_A d_B rvarepsilon2 log(d
• Score: 2.1198879079315573
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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)$ channel queries when $d_A= rd_B$, and $Ω\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A\le rd_B/2$. These lower bounds improve upon the best previous $Ω(d_A d_B r)$ bound by introducing explicit $\varepsilon$-dependence, and they are optimal up to logarithmic factors. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.

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