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

Thomas Wagner, Hermann Kampermann, Dagmar Bruß, and Martin Kliesch

Institut für Theoretische Physik, Heinrich-Heine-University Düsseldorf, Germany

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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.

Real quantum computers are sensitive to noise from the environment. A detailed description of this noise can help to mitigate it in many situations. However, learning such a description can be difficult and often requires many measurements. In this work, we combine ideas from the characterization of quantum systems and quantum error correction. We show that standard error correction schemes yield much information that is usually neglected. Under some conditions, using only the measurements performed during these schemes is already sufficient to obtain a detailed characterization of the noise. We rigorously derive these conditions and sketch a practical characterization scheme based on these ideas. Our approach suggests an additional avenue for the characterization of quantum devices. In particular, it reduces the required effort by making more efficient use of information that is measured anyway.

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Cited by

[1] Thomas Wagner, Hermann Kampermann, Dagmar Bruß, and Martin Kliesch, "Learning Logical Pauli Noise in Quantum Error Correction", Physical Review Letters 130 20, 200601 (2023).

[2] Robin Harper and Steven T. Flammia, "Learning Correlated Noise in a 39-Qubit Quantum Processor", PRX Quantum 4 4, 040311 (2023).

[3] Armands Strikis, Simon C. Benjamin, and Benjamin J. Brown, "Quantum Computing is Scalable on a Planar Array of Qubits with Fabrication Defects", Physical Review Applied 19 6, 064081 (2023).

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