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.