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Higher-group symmetry of (3+1)D fermionic $\mathbb{Z}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
by Maissam Barkeshli, Po-Shen Hsin, Ryohei Kobayashi
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Submission summary
| Authors (as registered SciPost users): | Po-Shen Hsin · Ryohei Kobayashi |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2311.05674v3 (pdf) |
| Date accepted: | 2024-04-24 |
| Date submitted: | 2024-04-10 03:37 |
| Submitted by: | Kobayashi, Ryohei |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
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.
Author comments upon resubmission
List of changes
1. We added footnote 4 clarifying the meaning of electric versus magnetic surface operator as requested in the first referee report.
2. We added footnote 5 for alternative explanation for the relation between Z8 and Z16 using cobordism group as requested in the second referee report.
3. We added reference to [72] for the lattice model of the domain wall with Majorana zero mode as requested in the second referee report.
4. We fixed a typo in Eq.99 for the Hamiltonian of the fermionic toric code.
Published as SciPost Phys. 16, 122 (2024)