Topological color code and symmetry-protected topological phases

Abstract

We study $(d-1)$-dimensional excitations in the $d$-dimensional color code that are created by transversal application of the $R_{d}$ phase operators on connected subregions of qubits. We find that such excitations are superpositions of electric charges and can be characterized by fixed-point wavefunctions of $(d-1)$-dimensional bosonic SPT phases with $(\mathbb{Z}_{2})^{\otimes d}$ symmetry. While these SPT excitations are localized on $(d-1)$-dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the non-triviality of emerging SPT wavefunctions. Moreover, these SPT-excitations can be physically realized as transparent gapped domain walls which exchange excitations in the color code. Namely, in the three-dimensional color code, the domain wall, associated with the transversal $R_{3}$ operator, exchanges a magnetic flux and a composite of a magnetic flux and loop-like SPT excitation, revealing rich possibilities of boundaries in higher-dimensional TQFTs. We also find that magnetic fluxes and loop-like SPT excitations exhibit non-trivial three-loop braiding statistics in three dimensions as a result of the fact that the $R_{3}$ phase operator belongs to the third-level of the Clifford hierarchy. We believe that the connection between SPT excitations, fault-tolerant logical gates and gapped domain walls, established in this paper, can be generalized to a large class of topological quantum codes and TQFTs.