Fault-Tolerant Quantum Error Correction for the universal D(S₃) topological code

The two-dimensional non-Abelian quantum double model D(S₃) offers a promising route toward fault-tolerant universal quantum computation, with experimental requirements comparable to the mainstream surface code. As shown by two of the present authors [Chen et al., npj Quantum Information 11, 112 (2025)], the D(S₃) code supports a fault-tolerant universal logical gate set, once it has been equipped with a robust decoder that is resilient to noisy syndrome measurements. This architecture eliminates the need for costly magic-state distillation, leading to a predictable reduction in overhead. However, constructing an efficient and fault-tolerant decoder remains an open challenge, due to the non-Abelian complexity of the code. Here, we introduce a space–time clustering (STC) decoder tailored to D(S₃); this uses the intrinsic fusion and braiding structure of the code to optimize its performance. Using tensor-network techniques, we perform large-scale numerical simulations under circuit-level Pauli noise and find thresholds comparable to those of the surface code. Our results demonstrate the practical feasibility of fault-tolerant universal quantum computation with D(S₃), establishing a concrete path toward scalable quantum processors with substantially reduced overhead.