Non-Clifford Gates via Non-Abelian Topological Orders

A critical mission for realizing large-scale quantum computing is the development of simple and resource-efficient implementations of logical non-Clifford gates. This talk discusses how topological field theory enables novel protocols for their implementation.

In the first part, based on arXiv:2510.20890, we present hybrid lattice surgery between Abelian and non-Abelian codes and show how it provides non-Clifford operations in the standard surface code, including non-Clifford gates and magic states, for qubits and qutrits.

In the second part, based on arXiv:2512.13777, we provide a purely 2D transversal realization of non-Clifford qubit phase gates at arbitrary levels of the Clifford hierarchy. We encode a logical qubit in the quantum double of a non-Abelian group on a triangular spatial patch and implement the logical gate transversally by stacking a symmetry-protected topological (SPT) phase onto the spatial region. This construction remains purely 2D and preserves locality and fault tolerance.