Higher-group symmetry in finite gauge theory and stabilizer codes

A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we derive the d-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in (d+1) space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the [O5]∈H5(BG,U(1)) obstruction than has appeared in previous work. We also show how the d-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We construct new logical gates in stabilizer codes using the d-group symmetry, such as the control-Z gate in (3+1)D Z2 toric code.