Magic Boundaries of 3D Color Codes

Abstract

We investigate boundaries of 3D color codes and provide a systematic classification into 101 distinct boundary types, including two novel classes. The first class consists of 1 boundary and is generated by sweeping the codimension-1 (2D) $T$-domain wall across the system and attaching it to the $X$-boundary that condenses only magnetic fluxes. Since the $T$-domain wall cannot condense on the $X$-boundary, a new $\textit{magic boundary}$ is produced, where the boundary stabilizers contain $XS$-stabilizers going beyond the conventional Pauli stabilizer formalism, and hence contains 'magic'. Neither electric nor magnetic excitations can condense on such a magic boundary, and only the composite of the magnetic flux and codimension-2 (1D) $S$-domain wall can condense on it, which makes the magic boundary going beyond the classification of the Lagrangian subgroup. The second class consists of 70 boundaries and is generated by sweeping the $S$-domain wall across a codimension-1 submanifold and attaching it to the boundary. This generates a codimension-2 (1D) $\textit{nested boundary}$ at the intersection. We also connect these novel boundaries to their previously discovered counterpart in the $\mathbb{Z}_2^3$ gauge theory, equivalent to three copies of 3D toric codes, where the $S$ and $T$ domain walls correspond to gauged symmetry-protected topological (SPT) defects. New boundaries are produced whenever the corresponding symmetry of the SPT defect remains unbroken on the boundary. Applications of the magic boundaries include implementing fault-tolerant non-Clifford logical gates, e.g., in the context of fractal topological codes.