Higher-group symmetry in finite gauge theory and stabilizer codes

Abstract

A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the dd-group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in (d+1)(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 [O_5] ∈ H^5(BG, U(1))[O5]∈H5(BG,U(1)) obstruction that has appeared in prior work. We also show how the dd-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the dd-group symmetry, such as a controlled Z gate in the (3+1) D \mathbb{Z}_2ℤ2 toric code.

Higher-group symmetry of (3+1)D fermionic $\mathbb{Z}_2$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry

“…We remark that the charge-flux attachment also follows from twisted dimensional reduction of the fermionic invertible phases on a circle with periodic boundary condition, similar to the reduction of bosonic SPT phases on a circle with nontrivial holonomy as discussed in [40]. The p+ip phase can be described by a single free Majorana fermion field theory in (2+1)D with a negative mass (where we choose a regulator such that fermion with positive mass belongs to the trivial phase).…”

“…The above automorphism can be derived by the charge-flux attachment argument [40]. Consider a magnetic flux defect intersecting the domain wall decorated with p+ip.…”

“…The 3-group symmetry structure between the 2-form symmetry and the 1-form symmetries, i.e. the junctions of surface defects that can produce the Wilson line, is discussed in [33,40,89]. Here we will focus on the three group structure from the junctions that involve the domain walls (or both the domain walls and the surface defects).…”

“…Recently, it has been observed that finite group gauge theories in (d +1)D generally possess a d-group symmetry [33,40], which arises by considering gauged invertible topological phases [41][42][43][44][45][46][47][48] decorated on lower dimensional submanifolds, in addition to the usual electric and magnetic defects. The interaction between the magnetic defects and the gauged invertible phases causes the higher form symmetry to mix into a non-trivial higher group.…”

“…The emergent higher symmetry operations corresponding to sweeping topological defects of varying codimension give rise to another class of fault-tolerant logical gates on the code subspace [33,[58][59][60][61]. This insight has led to new ways of implementing logical gates even in the well-studied 2 toric codes [40,62]. Remarkably, Ref.…”