Higher-group symmetry in finite gauge theory and stabilizer codes
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| Format: | Preprint |
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2022
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| _version_ | 1866917627560984576 |
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| author | Barkeshli, Maissam Chen, Yu-An Hsin, Po-Shen Kobayashi, Ryohei |
| author_facet | Barkeshli, Maissam Chen, Yu-An Hsin, Po-Shen Kobayashi, Ryohei |
| contents | 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 $d$-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 $[O_5] \in H^5(BG, U(1))$ obstruction that has appeared in prior 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 discover new logical gates in stabilizer codes using the $d$-group symmetry, such as a Controlled-Z gate in (3+1)D $\mathbb{Z}_2$ toric code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11764 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Higher-group symmetry in finite gauge theory and stabilizer codes Barkeshli, Maissam Chen, Yu-An Hsin, Po-Shen Kobayashi, Ryohei Strongly Correlated Electrons High Energy Physics - Theory Quantum Algebra Quantum Physics 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 $d$-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 $[O_5] \in H^5(BG, U(1))$ obstruction that has appeared in prior 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 discover new logical gates in stabilizer codes using the $d$-group symmetry, such as a Controlled-Z gate in (3+1)D $\mathbb{Z}_2$ toric code. |
| title | Higher-group symmetry in finite gauge theory and stabilizer codes |
| topic | Strongly Correlated Electrons High Energy Physics - Theory Quantum Algebra Quantum Physics |
| url | https://arxiv.org/abs/2211.11764 |