  The performance of quantum error correction can be significantly improved if
detailed information about the noise is available, allowing to optimize both
codes and decoders. It has been proposed to estimate error rates from the
syndrome measurements done anyway during quantum error correction. While these
measurements preserve the encoded quantum state, it is currently not clear how
much information about the noise can be extracted in this way. So far, apart
from the limit of vanishing error rates, rigorous results have only been
established for some specific codes. In this work, we rigorously resolve the
question for arbitrary stabilizer codes. The main result is that a stabilizer
code can be used to estimate Pauli channels with correlations across a number
of qubits given by the pure distance. This result does not rely on the limit of
vanishing error rates, and applies even if high weight errors occur frequently.
Moreover, it also allows for measurement errors within the framework of quantum
data-syndrome codes. Our proof combines Boolean Fourier analysis, combinatorics
and elementary algebraic geometry. It is our hope that this work opens up
interesting applications, such as the online adaptation of a decoder to
time-varying noise.
