Braiding via symmetry transformation: universal transversal gate set for topological codes

Recent studies of the relationship between symmetry and topology have stimulated a series of new discoveries. A recently developed theory of ``quantum origami" revealed the deep relation between mapping glass group, topological symmetry and transversal gates in fault-tolerant quantum computation (arXiv:1711.05752). Motivated by this finding, here we show that elements in the spherical braid group with n punctures (anyons) can be completely generated with two isometries related to different rotational symmetries of the punctured surface. By folding the surface to 2n layers connected with gapped boundaries, we acquire a ``folded fan" model for quantum computers in reminiscence of the ``pancake" model consisting of multiple disconnected layers. The two isometries in the unfolded manifold can be mapped into onsite symmetry in the folded fan, corresponding to fault-tolerant transversal logical gates which can be used to perform quantum computation. For the non-abelian Fibonacci-Turaev-Viro code, these transversal gates form a universal set, along with a constant-depth circuit switching the symmetry. In contrast to the code switching approach to circumvent the Eastin-Knill no-go theorem, we achieve the circumvention using a symmetry switching approach with constant time overhead.