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.

中文翻译：

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泡利通道可以从量子纠错中的校正子测量来估计

如果可以获得有关噪声的详细信息，则可以显着提高量子纠错的性能，从而优化代码和解码器。已经提议根据在量子纠错期间无论如何完成的校正子测量来估计错误率。虽然这些测量保留了编码的量子态，但目前尚不清楚通过这种方式可以提取多少关于噪声的信息。到目前为止，除了消失错误率的限制之外，仅针对某些特定代码建立了严格的结果。
在这项工作中，我们严格解决了任意稳定器代码的问题。主要结果是，稳定器代码可用于估计泡利通道，这些通道具有与纯距离给出的多个量子比特之间的相关性。此结果不依赖于消失错误率的限制，即使频繁发生高权重错误也适用。此外，它还允许在量子数据综合症代码的框架内出现测量误差。我们的证明结合了布尔傅立叶分析、组合学和初等代数几何。我们希望这项工作能够开辟有趣的应用，例如解码器对时变噪声的在线适应。