Improved Lower Bounds for Learning Quantum Channels in Diamond Distance

TL;DR

This paper establishes near-optimal lower bounds on the number of queries needed to learn quantum channels in diamond distance, improving previous bounds by explicitly incorporating the error parameter.

Contribution

It provides new lower bounds for quantum channel learning complexity that include explicit dependence on the error tolerance, refining previous bounds.

Findings

Lower bounds depend explicitly on the error parameter ε.

Bounds are tight up to logarithmic factors in certain regimes.

Constructs ensembles of channels with specific separation properties.

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 $ \Omega\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A= rd_B$, and $\Omega\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A\le rd_B/2$. These lower bounds improve upon the best previous $\Omega(d_A d_B r)$ bound by introducing explicit, near-optimal $\varepsilon$-dependence. Moreover, when $d_A\le rd_B/2$, 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.