It has recently been understood that the complete global symmetry of topological gauge theories contains the structure of a higher-group. In this talk, we look at the algebraic structure of symmetry in (3+1)D Z2 gauge theory with an emergent fermion, and point out that pumping chiral p+ip topological states gives rise to a Z8 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we discuss in detail. We then see that in the context of stabilizer quantum codes, one can obtain various logical non-Clifford gates of (3+1)D Z2 toric code (w/ emergent fermion) including the CCZ, CS and T gates, which are all implemented by pumping a p+ip topological state through a 3d space. The talk is mainly based on a recent work with Maissam Barkeshli and Po-Shen Hsin, which is found in https://arxiv.org/abs/2311.05674 .

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