Fault-tolerant mixed boundary punctures on the toric code

Defects on the toric code, a well-known exactly solvable Abelian anyon model, can exhibit non-Abelian statistical properties, which can be classified into punctures and twists. Benhemou et al. Phys. Rev. A 105 042417 (2022) introduced a mixed boundary puncture model that integrates the advantages of both punctures and twists. They proposed that non-Abelian properties could be realized in the symmetric subspace \\left|++\right\rangle,\left|-\right\rangle \. This work demonstrates that the nontrivial antisymmetric subspace \\left|+-\right\rangle,\left|-+\right\rangle \ also supports non-Abelian statistics. The mixed boundary puncture model is shown to be fault-tolerant in both subspaces, offering resistance to collective dephasing noise and collective rotation noise. In addition, we propose and validate a quantum information masking scheme within the three-partite mixed boundary puncture model.