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 $Obigl(frac{1}{epsilon^2 eta}bigr) log(n/epsilon)$ applications of the channel.
تحميل البحث