Fig. 4: Creation and braiding of CC defects and their relation to parafermions.

a Ground state of the \({{\mathbb{Z}}}_{3}\) toric code with a CC defect pair. The endpoints of the thick line, representing the CC construction circuit U, correspond to high-weight stabilizers A₀ and B₅ (defined in Supplementary Fig. 9c). These stabilizers are marked by a hatching pattern with a ' × ' symbol, and their values are omitted for clarity as they label the internal state of the defects which is not locally accessible. b A sketch of the braiding experiment in (c–f). A flux pair \(m-\bar{m}\) is created. \(\bar{m}\) is transmuted into m by commuting it through the CC defect line, and it is then fused with the fixed m anyon at the top right corner through a sequence of four steps (c–f), by applying \({{\mathcal{X}}}_{1}\), \({{\mathcal{X}}}_{2}^{\dagger }\), \({{\mathcal{X}}}_{3}^{\dagger }\), and \({{\mathcal{X}}}_{4}^{\dagger }\). g Outline of the braiding experiment in (h) where the CC defect pair is fused. This is achieved by applying the same circuit U used in (a) (see Supplementary Note 5). The dashed line shows the path (as implemented by U) taken to fuse the defect pair.The altered state of the CC defect pair is revealed as a flux m at one endpoint. i A sketch of the braiding experiment in (j). We prepare the ground state and create two parafermion defect pairs. The m flux from the pair \(m-\bar{m}\) created in the second plaquette from the top, leftmost corner remains fixed, while its partner \(\bar{m}\) anyon is commuted through two parafermion defect pairs. The resulting m is then fused with the pinned m to give a single \(\bar{m}\) anyon.