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.
 A. Robertson, C. Granade, S. D. Bartlett, and S. T. Flammia, Tailored codes for small quantum memories, Phys. Rev. Applied 8, 064004 (2017).
 E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002), arXiv:quant-ph/0110143 [quant-ph].
 S. T. Spitz, B. Tarasinski, C. W. J. Beenakker, and T. E. O'Brien, Adaptive weight estimator for quantum error correction in a time-dependent environment, Advanced Quantum Technologies 1, 1870015 (2018).
 Z. Babar, P. Botsinis, D. Alanis, S. X. Ng, and L. Hanzo, Fifteen years of quantum LDPC coding and improved decoding strategies, IEEE Access 3, 2492 (2015).
 S. Huang, M. Newman, and K. R. Brown, Fault-tolerant weighted union-find decoding on the toric code, Physical Review A 102, 10.1103/physreva.102.012419 (2020).
 A. S. Darmawan and D. Poulin, Linear-time general decoding algorithm for the surface code, Physical Review E 97, 10.1103/physreve.97.051302 (2018).
 M. Ware, G. Ribeill, D. Ristè, C. A. Ryan, B. Johnson, and M. P. da Silva, Experimental Pauli-frame randomization on a superconducting qubit, Phys. Rev. A 103, 042604 (2021).
 S. J. Beale, J. J. Wallman, M. Gutiérrez, K. R. Brown, and R. Laflamme, Quantum error correction decoheres noise, Phys. Rev. Lett. 121, 190501 (2018).
 A. G. Fowler, D. Sank, J. Kelly, R. Barends, and J. M. Martinis, Scalable extraction of error models from the output of error detection circuits (2014).
 M.-X. Huo and Y. Li, Learning time-dependent noise to reduce logical errors: real time error rate estimation in quantum error correction, New J. Phys. 19, 123032 (2017).
 J. Combes, C. Ferrie, C. Cesare, M. Tiersch, G. J. Milburn, H. J. Briegel, and C. M. Caves, In-situ characterization of quantum devices with error correction (2014).
 T. Wagner, H. Kampermann, D. Bruß, and M. Kliesch, Optimal noise estimation from syndrome statistics of quantum codes, Phys. Rev. Research 3, 013292 (2021).
 J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Lucero, M. Neeley, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, and J. M. Martinis, Scalable in situ qubit calibration during repetitive error detection, Phys. Rev. A 94, 032321 (2016).
 Y. Fujiwara, Ability of stabilizer quantum error correction to protect itself from its own imperfection, Phys. Rev. A 90, 062304 (2014), arXiv:1409.2559 [quant-ph].
 N. Delfosse, B. W. Reichardt, and K. M. Svore, Beyond single-shot fault-tolerant quantum error correction, IEEE Transactions on Information Theory 68, 287 (2022).
 A. Zia, J. P. Reilly, and S. Shirani, Distributed parameter estimation with side information: A factor graph approach, in 2007 IEEE International Symposium on Information Theory (2007) pp. 2556–2560.
 D. Koller and N. Friedman, Probabilistic Graphical Models: Principles and Techniques - Adaptive Computation and Machine Learning (The MIT Press, 2009).
 T. Chen and LiTien-Yien, Solutions to systems of binomial equations, Annales Mathematicae Silesianae 28, 7 (2014).
 B. M. Varbanov, F. Battistel, B. M. Tarasinski, V. P. Ostroukh, T. E. O'Brien, L. DiCarlo, and B. M. Terhal, Leakage detection for a transmon-based surface code, NPJ Quantum Inf. 6, 10.1038/s41534-020-00330-w (2020).
 Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, and Peter Zoller, "The randomized measurement toolbox", arXiv:2203.11374.
 Armands Strikis, Simon C. Benjamin, and Benjamin J. Brown, "Quantum computing is scalable on a planar array of qubits with fabrication defects", arXiv:2111.06432.
 Thomas Wagner, Hermann Kampermann, Dagmar Bruß, and Martin Kliesch, "Learning logical quantum noise in quantum error correction", arXiv:2209.09267.
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