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Higher-group symmetry in finite gauge theory and stabilizer codes

by Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin, Ryohei Kobayashi

Submission summary

Authors (as registered SciPost users): Yu-An Chen · Po-Shen Hsin · Ryohei Kobayashi
Submission information
Preprint Link: scipost_202307_00028v1  (pdf)
Date submitted: 2023-07-21 20:56
Submitted by: Kobayashi, Ryohei
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


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.

Current status:
In refereeing

Submission & Refereeing History

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Submission scipost_202307_00028v1 on 21 July 2023

Reports on this Submission

Anonymous Report 1 on 2023-9-22 (Invited Report)


-detailed study of higher-group symmetry in finite gauge theories and several exciting applications, such as the classification of fermionic SPTs and fault-tolerant gates
-very well written and clear


-outlook with possible future directions could be mentioned


This paper comprehensively studies the higher group symmetry and its 't Hooft anomaly in finite gauge theories. It describes the higher-group symmetry as arising from a generalization of the Witten effect and the charge-flux attachment. It provides several explicit examples, including field theories and lattice models. The applications discussed in the paper are exciting. For instance, studying the fault-tolerant logical gates from this perspective could lead to insights for practical applications.

The paper's presentation and results are excellent. I wholeheartedly recommend publishing it in SciPost Physics.

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: excellent

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