Protected gates for topological quantum field theories (Prof. R. König)

For generic anyon models, we show that such unitaries only generate a finite group, and hence do not provide universality. For non-abelian models, we find that such automorphisms are very limited: for example, there is no non-trivial gate for Fibonacci anyons. More generally, systems with computationally universal braiding have no such gates. For Ising anyons, protected gates are elements of the Pauli group.

These results are derived by relating such automorphisms to symmetries of the underlying anyon model: protected gates realize automorphisms of the Verlinde algebra. We additionally use the compatibility with basis changes to characterize the logical action.

This is joint work with M. Beverland, O. Buerschaper, F. Pastawski, J. Preskill and S. Sijher.