Gapped boundaries, group cohomology and fault-tolerant logical gates

This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the $d$-dimensional quantum double model by using $d$-cocycles functions ($d\geq 2$). We point out that the system supports $m$-dimensional excitations ($m