Gauge theories are important descriptions for many physical phenomena and systems in quantum computation. Automorphism of gauge group naturally gives global symmetries of gauge theories. In this work we study such symmetries in gauge theories induced by automorphisms of the gauge group, when the gauge theories have nontrivial topological actions in different spacetime dimensions. We discover the automorphism symmetry can be extended, become a higher group symmetry, and/or become a non-invertible symmetry. We illustrate the discussion with various models in field theory and on the lattice. In particular, we use automorphism symmetry to construct new transversal non-Clifford logical gates in topological quantum codes. In particular, we show that 2+1d $\mathbb{Z}_N$ qudit Clifford stabilizer models can implement non-Clifford transversal logical gate in the 4th level $\mathbb{Z}_N$ qudit Clifford hierarchy for $N\geq 3$ , extending the generalized Bravyi-König bound proposed in the companion paper [arXiv:2511.02900] for qubits.Gauge theories are important descriptions for many physical phenomena and systems in quantum computation. Automorphism of gauge group naturally gives global symmetries of gauge theories. In this work we study such symmetries in gauge theories induced by automorphisms of the gauge group, when the gauge theories have nontrivial topological actions in different spacetime dimensions. We discover the automorphism symmetry can be extended, become a higher group symmetry, and/or become a non-invertible symmetry. We illustrate the discussion with various models in field theory and on the lattice. In particular, we use automorphism symmetry to construct new transversal non-Clifford logical gates in topological quantum codes. In particular, we show that 2+1d $\mathbb{Z}_N$ qudit Clifford stabilizer models can implement non-Clifford transversal logical gate in the 4th level $\mathbb{Z}_N$ qudit Clifford hierarchy for $N\geq 3$, extending the generalized Bravyi-König bound proposed in the companion paper [arXiv:2511.02900] for qubits.

cond-mat.str-elCross-listedarXiv:2511.15783SciPost PhysicsNov 21, 20250 citations

Higher-group symmetry of (3+1)D fermionic $\mathbb{Z}_2$ gauge theory: Logical CCZ, CS, and T gates from higher symmetryHigher-group symmetry of (3+1)D fermionic $\mathbb{Z}_2$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry

Maissam Barkeshli, Po-Shen Hsin, Ryohei Kobayashi

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 ℤ 2 gauge theory with an emergent fermion, and point out that pumping chiral p+ip p + i p topological states gives rise to a \mathbb{Z}_{8} 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 T 3 (3-torus) and T^2 \rtimes_{C_2} S^1 T 2 ⋊ C 2 S 1 (2-torus bundle over the circle) respectively, and pumping p+ip p + i p states. Our considerations also imply the possibility of a logical T T gate by placing the code on \mathbb{RP}^3 ℝ ℙ 3 and pumping a p+ip p + i p topological state.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 ℤ 2 gauge theory with an emergent fermion, and point out that pumping chiral p+ip p + i p topological states gives rise to a \mathbb{Z}_{8} 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 T 3 (3-torus) and T^2 \rtimes_{C_2} S^1 T 2 ⋊ C 2 S 1 (2-torus bundle over the circle) respectively, and pumping p+ip p + i p states. Our considerations also imply the possibility of a logical T T gate by placing the code on \mathbb{RP}^3 ℝ ℙ 3 and pumping a p+ip p + i p topological state.

arXiv:2311.05674SciPost PhysicsMay 7, 202418 citations

On the Structure of Quantum Logic

Peter D. Finch

Springer eBooksJan 1, 197553 citations

\mathbb{Z}_N

qudit Clifford stabilizer models can implement non-Clifford transversal logical gate in the 4th level

\mathbb{Z}_N

qudit Clifford hierarchy for

N\geq 3

, extending the generalized Bravyi-König bound proposed in the companion paper [arXiv:2511.02900] for qubits.

Higher-group symmetry of (3+1)D fermionic

\mathbb{Z}_2

gauge theory: Logical CCZ, CS, and T gates from higher symmetry