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 є in l∞ using just O (1/ є ² ) log( n/ є ) 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/ є) 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 ≤ 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 − n. In the regime of small n we extend our algorithm to achieve multiplicative precision 1 ± є (i.e., additive precision єn) using just O(1 / є²n) log (n/ є ) applications of the channel.