Topological color code and SPT phases
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 (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.
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