Integer Linear Programming Decoder for Abelian and Non-Abelian Topological Codes

Topological orders serve as robust quantum error-correcting codes where anyon excitations act as error syndromes. However, existing decoders either are restricted to simple particle-antiparticle matching in Abelian codes or rely on clustering methods that neglect anyon data and fusion properties in non-Abelian codes. This paper asks whether a unified, general-purpose decoder can be constructed for arbitrary topological orders by leveraging classical optimization techniques.

The authors introduce an integer linear programming (ILP) framework for minimum-weight decoding across arbitrary topological codes. The method linearizes the error-correction problem by introducing auxiliary decision variables and encoding anyon fusion rules, as well as noise correlations between different species, as linear constraints. Classical optimization then identifies the most probable error configuration consistent with measured syndromes. The approach is benchmarked on three specific topological orders: the Abelian Z2 toric code under depolarizing noise with correlated errors, the Abelian Z3 topological code without a pairwise matching decoder, and the non-Abelian D4 topological code.

For the Abelian Z2 topological code under depolarizing noise, the ILP decoder achieves an error-correction threshold of 18.039%, significantly outperforming uncorrelated minimum-weight perfect matching and most existing decoders while closely approaching the theoretical optimal limit. For the Abelian Z3 topological order, the decoder achieves a threshold of 15.346%, substantially exceeding prior renormalization group methods. Finally, for the non-Abelian D4 topological order under noise generating all anyon species, the ILP decoder demonstrates a clear performance advantage over matching-based alternatives, establishing the method as a flexible and powerful solution for both Abelian and non-Abelian systems.