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The Surface Code beyond Pauli Channels: Logical Noise Coherence, Information-Theoretic Measures, and Errorfield-Double Phenomenology

Jan Behrends1,* and Benjamin Béri1,2,†

  • *Contact author: jan.behrends@yahoo.de
  • †Contact author: bfb26@cam.ac.uk
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PRX Quantum 6, 040350 – Published 2 December, 2025

DOI: https://doi.org/10.1103/psf5-b6j2

Abstract

We consider the surface code under errors featuring both coherent and incoherent components and study the coherence of the corresponding logical noise channel and how this impacts information-theoretic measures of code performance, namely coherent information and quantum relative entropy. Using numerical simulations and developing a phenomenological field theory, focusing on the most general single-qubit X-error channel, we show that, for any nonzero incoherent noise component, the coherence of the logical noise is exponentially suppressed with the code distance. We also find that the information-theoretic measures require this suppression to detect optimal thresholds for Pauli recovery; for this they thus require increasingly large distances for increasing error coherence and ultimately break down for fully coherent errors. To obtain our results, we develop a statistical mechanics mapping and a corresponding matrix-product-state algorithm for approximate syndrome sampling. These methods enable the large-scale simulation of these non-Pauli errors, including their maximum-likelihood thresholds, away from the limits captured by previous approaches.

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Popular Summary

The fragility of quantum states limits the potential of quantum computers since interactions with the environment or imperfect control introduce errors that can rapidly destroy quantum information. Quantum error correction (QEC) offers a potential solution to this challenge. Most studies of QEC performance assume purely random, probabilistic errors and assess corresponding logical channels with information-theoretic measures. However, physical devices often suffer from coherent errors: systematic imperfections such as consistent over-rotations in a qubit’s control pulses, whose effect on the resulting logical noise is unknown.

In this work, the authors explore the surface code, a leading approach for QEC, under generic single-qubit errors that combine incoherent (probabilistic) and coherent (unitary) components. By mapping the decoding problem to a statistical mechanics model and an efficient sampling method employing matrix-product states, they numerically simulate large surface codes, compute QEC thresholds, and develop an effective field-theoretic description of the code under such errors. For even small incoherent noise, they find that the coherence of the logical noise decays exponentially with code distance—larger codes “wash out” logical noise coherence. This is critical for information-theoretic measures, which, as the authors show, tacitly rely on low logical noise coherence to assess QEC performance.

These results demonstrate that even modest levels of incoherent noise suppress unwanted coherence at the logical level, reinforcing the viability of surface codes in realistic settings. However, the breakdown of standard information-theoretic measures at high coherence suggests caution when using them to assess small devices with large coherent-error components. Future work could extend this framework to more general single- and multi-qubit errors and circuit-level noise.

Article Text

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PRX Quantum 6, 040350– Published 2 December, 2025
  • Received 2 February 2025
  • Revised 30 August 2025
  • Accepted 21 October 2025
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