Pauli error estimation via Population Recovery

Motivated by estimation of quantum noise models, we study the problem learning a Pauli channel, or more generally error rates an arbitrary channel. By employing novel reduction to "Population Recovery" problem, give extremely simple algorithm that learns $n$-qubit channel precision $\epsilon$ in $\ell_\infty$ using just $O(1/\epsilon^2) \log(n/\epsilon)$ applications This is optimal up logarithmic factors. Our uses only unentangled state preparation and measurements, post-measurement classical runtime $O(1/\epsilon)$ factor larger than measurement data size. It also impervious limited model where heralded failures occur independently with probability $\le 1/4$. We then consider case close identity, meaning no-error outcome occurs $1-\eta$. In regime small $\eta$ extend our achieve multiplicative $1 \pm \epsilon$ (i.e., additive $\epsilon \eta$) $O\bigl(\frac{1}{\epsilon^2 \eta}\bigr)