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  • Open Access

Statistical Mechanical Mapping and Maximum-Likelihood Thresholds for the Surface Code under Generic Single-Qubit Coherent Errors

Jan Behrends1,* and Benjamin Béri1,2

  • *Contact author: jan.behrends@yahoo.de
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PRX Quantum 6, 040305 – Published 7 October, 2025

DOI: https://doi.org/10.1103/gskb-t5ql

Abstract

The surface code, one of the leading candidates for quantum error correction, is known to protect encoded quantum information against stochastic, i.e., incoherent errors. The protection against coherent errors, such as from unwanted gate rotations, is however understood only for special cases, such as rotations about the X or Z axes. Here we consider generic single-qubit coherent errors in the surface code, i.e., rotations by angle α about an axis that can be chosen arbitrarily. We develop a statistical mechanical mapping for such errors and perform entanglement analysis in transfer matrix space to numerically establish the existence of an error-correcting phase, which we chart in a subspace of rotation axes to estimate the corresponding maximum-likelihood thresholds αth. The classical statistical mechanics model we derive is a random-bond Ising model with complex couplings and four-spin interactions (i.e., a complex-coupled Ashkin-Teller model). The error-correcting phase, α<αth, where the logical error rate decreases exponentially with code distance, is shown to correspond in transfer matrix space to a gapped one-dimensional quantum Hamiltonian exhibiting spontaneous breaking of a Z2 symmetry. Our numerical results rest on two key ingredients: (i) we show that the state evolution under the transfer matrix, a nonunitary (1+1)-dimensional quantum circuit, can be efficiently numerically simulated using matrix product states; and (ii) based on this approach, we also develop an algorithm to (approximately) sample syndromes based on their Born probability. The αth values we find show that the maximum-likelihood thresholds for coherent errors are larger than those for the corresponding incoherent errors (from the Pauli twirl), and significantly exceed the values found using minimum weight perfect matching.

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Popular Summary

Quantum computers rely on error-correcting codes to protect quantum information. The performance of such codes, the most prominent of which is the surface code, is typically studied while assuming random (incoherent) errors. However, real devices often suffer from coherent errors (like small but systematic rotations). How much coherent noise can the surface code tolerate?

To study this, the authors translate the question, for single-qubit coherent errors, into an unusual statistical mechanics problem, which they describe in terms of the quantum dynamics of a lower-dimensional system. Using this approach, they then develop an efficient computational technique, including for the hard problem of sampling syndromes (error-detecting measurements), with which they explore the surface code’s performance. They find that the fundamental limits of error correction (the thresholds for maximum-likelihood decoding to infer and correct errors) correspond to surprisingly high coherent-error tolerance.

These results pave the way for more efficient quantum error correction in real hardware, where coherent errors can be prevalent. By quantifying the thresholds and providing a simulation framework, the study may inspire new decoders and facilitate the more efficient use of fault-tolerant quantum systems. As a next step, extending these techniques to a broader set of errors and codes could bring us closer to robust, fault-tolerant quantum computing.

Article Text

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PRX Quantum 6, 040305– Published 7 October, 2025
  • Received 4 December 2024
  • Revised 27 May 2025
  • Accepted 28 July 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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