Automorphism in Gauge Theory and Clifford-Hierarchy Stabilizer Codes

For Pauli stabilizer codes—such as surface codes and color codes—transversal logical gates are strongly constrained by the Bravyi–König bound. In particular, implementing a non-Clifford gate transversally (as required for universality) demands three spatial dimensions. In this talk, I will explain how these constraints can be overcome by moving beyond Pauli stabilizers to a broader class of non-Pauli stabilizer codes, dubbed a Clifford hierarchy stabilizer codes.

I will present constructions showing that certain 2D Clifford stabilizer codes admit a transversal T gate, while 3D non-Clifford stabilizer codes can admit a transversal sqrt{T} gate. These transversal gates correspond to automorphism symmetries of twisted gauge theory, and the code realizes non-Abelian topological order.