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

TL;DR

This paper explores how automorphisms of gauge groups induce higher and non-invertible symmetries in gauge theories, enabling the construction of new transversal non-Clifford logical gates in topological quantum codes.

Contribution

It demonstrates the extension of automorphism symmetries into higher and non-invertible forms and applies this to develop non-Clifford gates in topological quantum codes.

Findings

Automorphism symmetries can be extended to higher group and non-invertible symmetries.

Constructed new transversal non-Clifford logical gates in 2+1d $ ext{Z}_N$ qudit models.

Extended the Bravyi-König bound for non-Clifford gates in topological quantum codes.

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-K\"onig bound proposed in the companion paper [arXiv:2511.02900] for qubits.