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

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