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 $\mathbb{Z}_2$ 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} $\mathbb{Z}_2$ 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 $\mathrm{SO}(n)$ nonlinear sigma model in the replica limit $n\to1$. 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 $\mathbb{Z}_{2}$ paramagnet. We probe this "non-classical" character through the output state's sensitivity to well-separated $\mathbb{Z}_{2}$ fluxes inserted in the initial state, which reveals persistent coherence between distinct $\mathbb{Z}_{2}$ charge trajectories at long distances.

The paper asks whether interference between coherent error strings can survive at long distances, producing recovery regimes and mixed-state order absent under stochastic Pauli noise. It studies a triangular-lattice surface code under uniform single-qubit $Z$ rotations followed by syndrome measurement, and its Wegner-dual Ising paramagnet under symmetric $ZZ$ rotations followed by $X$ dephasing. A replica description yields an effective $\mathrm{SO}(n)$ nonlinear sigma model at $n\to1$, with a stable weak-coupling critical phase. Above a threshold $\theta_c/\pi\simeq0.0882\pm0.0005$, the code is partially decodable: at fixed cylinder aspect ratio $\kappa=T/L$, the Bayesian Pauli-decoder fidelity approaches an angle-independent, geometry-dependent limit rather than unity. Analytic replica resummation predicts $1-\mathcal F_*\sim e^{-\pi\kappa/16}/\sqrt{\kappa}$ at large $\kappa$. In the dual system, twist-field correlations predict algebraic strong-to-weak symmetry breaking, with fidelity correlator $C_F(r)\sim r^{-1/16}$ and generalized Chernoff correlators scaling as $r^{-k(1-k)/4}$. Born-rule-sampled simulations test these predictions; network-model conductivity grows as $(2\pi)^{-1}\log L$, with transition exponent estimates $\nu\approx2.3$–$2.5$. A flux-insertion distinguishability probe diagnoses persistent interference and obstructs an effective Pauli-dephasing description of the channel. Limitations include uncomputed logarithmic corrections, large crossover lengths near the threshold and $\theta=\pi/4$, and a fitted nonuniversal boundary prefactor in the fidelity formula, whose twist approximation is controlled at large aspect ratio. Recovery is restricted to Bayesian Pauli corrections, not unrestricted optimal recovery; maximum-likelihood decoding shares its threshold but not necessarily its fidelity.