Automorphism in Gauge Theories: Higher Symmetries and Transversal Non-Clifford Logical Gates

TL;DR

This work analyzes how automorphisms of gauge groups in twisted and untwisted gauge theories generate a rich web of symmetries, including invertible, extended, higher-group, and non-invertible structures via decorations by gauged SPT defects. It provides concrete lattice and field-theoretic constructions showing how these automorphisms can implement transversal non-Clifford gates, notably yielding a 4th-level Clifford-hierarchy transversal gate for $\mathbb{Z}_N$ qudits in 2+1d, and a transversal CS gate in 3+1d $\mathbb{Z}_2\times\mathbb{Z}_2$ toric code. The results extend known Bravyi-König-type bounds for qubits to qudit systems and reveal how higher-group symmetries arise from interactions between automorphisms and gauged SPT sectors across dimensions. These insights deepen the connection between gauge theory symmetry structures and fault-tolerant quantum computation, with implications for phase structure, anomalies, and code design. Practical impact includes new avenues for universal quantum computation using topological codes and a framework for analyzing symmetry-enabled logical gates in a broad class of topological phases.

Abstract

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.

Figures (6)

• Figure 1: Automorphism symmetry $\rho:G\rightarrow G$ in twisted gauge theory with topological action $\omega$ needs to decorate with gauged SPT symmetry $e^{i\int \alpha}$ for automorphism $\rho$ that preserves the cohomology class $[\omega]$ but in general not the cocycle, $\rho^*\omega=\omega+d\alpha$. The time direction goes from the right to the left, and the interface is oriented.
• Figure 2: Automorphism symmetry can be extended by the gauged SPT symmetry $e^{i\int \eta}$ due to the decoration by gauged SPT defect $U_\rho=e^{i\int \alpha_\rho}V_\rho$.
• Figure 3: Sandwich construction for automorphism symmetry in twisted gauge theory with gauge group $G$. The two blue interfaces break the gauge group from $G$ to subgroup $K$, such that the automorphism symmetry becomes invertible higher-group symmetry in gauge group $K$. As a whole sandwich, the automorphism symmetry is non-invertible.
• Figure 4: Junction of automorphism symmetry (black) in untwisted gauge theory with gauged SPT symmetry (red) for cocycle $\omega$ of the gauge group. For automorphism that changes $\omega$ by $d\alpha$, the junction (blue) is decorated with $e^{i\int\alpha}$.
• Figure 5: Fusing automorphism symmetry defects in the presence of a gauged SPT symmetry defect $\omega$ in $k$ dimensions. This leaves a $(k-1)$-dimensional gauged SPT symmetry $\eta$, regarded as a higher-group symmetry.

Theorems & Definitions (5)

• Theorem 5.1
• Theorem 5.2
• Theorem 5.3
• Theorem 5.4
• proof