Pauli error estimation via Population Recovery

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