Higher-order topological superconductors as generators of quantum codes

Abstract

We show that interactions can drive a class of higher order topological superconductors (HOTSCs) into symmetry-enriched topologically ordered phases exemplified by topological quantum error correcting codes. In two dimensions, interacting HOTSCs realize various topologically ordered surface and color codes. In three dimensions, interactions can drive HOTSCs protected by subsystem symmetries into recently discovered fracton phases. We explicitly relate fermion parity operators underlying the gapless excitations of the HOTSC to the Wilson algebra of symmetry-enriched quantum codes. Arrays of crossed Majorana wires provide an experimental platform for realizing fracton matter and for probing the quantum phase transition between HOTSCs and the topologically ordered phase.