Algebraic strong-to-weak symmetry breaking and partial decodability in the surface code
Abstract
Coherent noise can give rise to unconventional regimes of quantum information recovery and mixed-state order. We study these phenomena in two settings related by Wegner duality: the triangular-lattice surface code subject to single-qubit Pauli-Z rotations followed by syndrome measurements, and a ℤ2 paramagnet subject to strongly symmetric decoherence. Increasing the rotation angle drives a transition from a decodable phase to a \emph{partially decodable} critical phase in the code, and from a strongly symmetric phase to a phase with \emph{algebraic} ℤ2 strong-to-weak spontaneous symmetry breaking (SWSSB) in the dual paramagnet. We develop a common description of these phenomena in terms of a weakly coupled SO(n) nonlinear sigma model in the replica limit n→1. Taking this replica limit analytically, we derive universal signatures of the critical phase: an optimal Pauli decoding fidelity that approaches a universal function of the system's aspect ratio in the thermodynamic limit, and a dual fidelity correlator that decays as a universal power law. These observables relate to correlations of twist operators in the sigma model description. Large-scale numerical simulations confirm these analytical predictions. We also argue that the algebraic SWSSB phase cannot be realized by any Pauli dephasing channel acting on the ℤ2 paramagnet. We probe this "non-classical" character through the output state's sensitivity to well-separated ℤ2 fluxes inserted in the initial state, which reveals persistent coherence between distinct ℤ2 charge trajectories at long distances.
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BibTeX
@misc{yan2026algebraic,
title = {{Algebraic strong-to-weak symmetry breaking and partial decodability in the surface code}},
author = {Stephen W. Yan and Yimu Bao and Sagar Vijay},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40567},
eprint = {2610.40567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40567},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebraic strong-to-weak symmetry breaking and partial decodability in the surface code
AU - Stephen W. Yan
AU - Yimu Bao
AU - Sagar Vijay
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40567
UR - https://arxiv.org/abs/2610.40567
AB - Coherent noise can give rise to unconventional regimes of quantum information recovery and mixed-state order. We study these phenomena in two settings related by Wegner duality: the triangular-lattice surface code subject to single-qubit Pauli-Z rotations followed by syndrome measurements, and a ℤ_2 paramagnet subject to strongly symmetric decoherence. Increasing the rotation angle drives a transition from a decodable phase to a \emph{partially decodable} critical phase in the code, and from a strongly symmetric phase to a phase with \emph{algebraic} ℤ_2 strong-to-weak spontaneous symmetry breaking (SWSSB) in the dual paramagnet. We develop a common description of these phenomena in terms of a weakly coupled SO(n) nonlinear sigma model in the replica limit n→1. Taking this replica limit analytically, we derive universal signatures of the critical phase: an optimal Pauli decoding fidelity that approaches a universal function of the system's aspect ratio in the thermodynamic limit, and a dual fidelity correlator that decays as a universal power law. These observables relate to correlations of twist operators in the sigma model description. Large-scale numerical simulations confirm these analytical predictions. We also argue that the algebraic SWSSB phase cannot be realized by any Pauli dephasing channel acting on the ℤ_2 paramagnet. We probe this "non-classical" character through the output state's sensitivity to well-separated ℤ_2 fluxes inserted in the initial state, which reveals persistent coherence between distinct ℤ_2 charge trajectories at long distances.
ER -