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This rigorous study establishes mathematically proven recovery thresholds for a broad class of two-dimensional topological codes under arbitrary local noise channels. The research confirms that logical information remains recoverable up to specific noise strengths while demonstrating the persistence of mixed-state topological order, including long-range entanglement and emergent higher-form symmetries. These findings provide critical theoretical bounds essential for advancing fault-tolerant quantum computing architectures beyond standard Pauli error models.
Mixed-state topological order and error-correction thresholds in non-Abelian codes: rigorous results
arXiv:2607.21706v1 Announce Type: new Abstract: We present a versatile and mathematically rigorous technique for bounding recovery thresholds in topological codes subject to noise. Our method captures the effect of applying an arbitrary (possibly non-Pauli) local noise channel to the code state of a broad class of two-dimensional codes, including surface codes, non-Abelian quantum doubles, and string-net codes. In each case, we prove that for noise strengths up to some explicit constant value, any initially encoded logical information can be recovered to high precision, and that the noise-corrupted state exhibits key hallmarks of mixed-state topological order: long-range entanglement and emergent higher-form symmetries. We also describe how these methods can be adapted to higher dimensions and correlated noise models.
Abstract:We present a versatile and mathematically rigorous technique for bounding recovery thresholds in topological codes subject to noise. Our method captures the effect of applying an arbitrary (possibly non-Pauli) local noise channel to the code state of a broad class of two-dimensional codes, including surface codes, non-Abelian quantum doubles, and string-net codes. In each case, we prove that for noise strengths up to some explicit constant value, any initially encoded logical information can be recovered to high precision, and that the noise-corrupted state exhibits key hallmarks of mixed-state topological order: long-range entanglement and emergent higher-form symmetries. We also describe how these methods can be adapted to higher dimensions and correlated noise models.
24 pages of main text, 40 pages of appendices. 8 figures
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el)
arXiv:2607.21706 [quant-ph]
(or arXiv:2607.21706v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2607.21706 arXiv-issued DOI via DataCite (pending registration)
From: Sun Woo P. Kim [view email] [v1] Thu, 23 Jul 2026 18:00:03 UTC (475 KB)
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