TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.16.5.122
TI  - Higher-group symmetry of (3+1)D fermionic $\mathbb{Z}_2$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry
PY  - 2024/05/07
UR  - https://scipost.org/SciPostPhys.16.5.122
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 16
IS  - 5
SP  - 122
A1  - Barkeshli, Maissam
AU  - Hsin, Po-Shen
AU  - Kobayashi, Ryohei
AB  - It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D $\mathbb{Z}_2$ gauge theory with an emergent fermion, and point out that pumping chiral $p+ip$ topological states gives rise to a $\mathbb{Z}_{8}$ 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization of $T^3$ (3-torus) and $T^2 \rtimes_{C_2} S^1$ (2-torus bundle over the circle) respectively, and pumping $p+ip$ states. Our considerations also imply the possibility of a logical $T$ gate by placing the code on $\mathbb{RP}^3$ and pumping a $p+ip$ topological state.
ER  -

@Article{10.21468/SciPostPhys.16.5.122,
	title={{Higher-group symmetry of (3+1)D fermionic $\mathbb{Z}_2$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry}},
	author={Maissam Barkeshli and Po-Shen Hsin and Ryohei Kobayashi},
	journal={SciPost Phys.},
	volume={16},
	pages={122},
	year={2024},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.16.5.122},
	url={https://scipost.org/10.21468/SciPostPhys.16.5.122},
}