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The performance of quantum error correction can be significantly improved ifdetailed information about the noise is available, allowing to optimize bothcodes and decoders. It has been proposed to estimate error rates from thesyndrome measurements done anyway during quantum error correction. While thesemeasurements preserve the encoded quantum state, it is currently not clear howmuch information about the noise can be extracted in this way. So far, apartfrom the limit of vanishing error rates, rigorous results have only beenestablished for some specific codes. In this work, we rigorously resolve thequestion for arbitrary stabilizer codes. The main result is that a stabilizercode can be used to estimate Pauli channels with correlations across a numberof qubits given by the pure distance. This result does not rely on the limit ofvanishing error rates, and applies even if high weight errors occur frequently.Moreover, it also allows for measurement errors within the framework of quantumdata-syndrome codes. Our proof combines Boolean Fourier analysis, combinatoricsand elementary algebraic geometry. It is our hope that this work opens upinteresting applications, such as the online adaptation of a decoder totime-varying noise.

Pauli channels can be estimated from syndrome measurements in quantum error correction