Decoherence and Wave-Function Deformation of $D_4$ Non-Abelian Topological Order

The effect of decoherence on topological order (TO) has been most deeply understood for the toric code, the paragon of Abelian TOs. We show that certain non-Abelian TOs can be analyzed and understood to a similar degree, despite being significantly richer. We consider both wave-function deformations and quantum channels acting on $D_4$ TO, which has recently been realized on a quantum processor. By identifying the corresponding local statistical mechanical spin or rotor model with $D_4$ symmetry, we find a remarkable stability against proliferating non-Abelian anyons. This is shown by leveraging a reformulation in terms of the tractable O(2) loop model in the pure state case and $n$ coupled O(2) loop models for Rényi-$n$ quantities in the decoherence case—corresponding to worldlines of the proliferating anyon with quantum dimension 2. In particular, we find that the purity $(n = 2)$ remains deep in the $D_4$ TO for any decoherence strength, while the $n → \infty$ limit becomes critical upon maximally decohering a particular anyon type, similar to our wave-function deformation result. The information-theoretic threshold $(n → 1)$ appears to be controlled by a disordered version of these statistical mechanical models, akin to the toric code case although significantly more robust. We furthermore use Monte Carlo simulations to explore the phase diagrams when multiple anyon types proliferate at the same time, leading to a continued stability of the $D_4$ TO in addition to critical phases with emergent U(1) symmetry. Instead of loop models, these are now described by net models corresponding to different anyon types coupled together according to fusion rules. This opens up the exploration of statistical mechanical models for decohered non-Abelian TO, which can inform optimal decoders and which in an ungauged formulation provides examples of non-Abelian strong-to-weak symmetry breaking.