
Protected gates for topological quantum field theories  Michael Beverland (Caltech)
 ABC Physics (Atomic/Bio/Condensed Matter) Seminar
Date: Friday, February 26, 2016 11:00 AM Location: PAT C520
Abstract
We study restrictions on localitypreserving unitary logical gates for topological quantum codes in two spatial dimensions. A localitypreserving operation is one which maps local operators to local operators  for example, a constantdepth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometricallylocal boundedstrength Hamiltonian. Localitypreserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of twodimensional topological field theories, we find that the localitypreserving logical gates are severely limited for codes which admit nonabelian anyons; in particular, there are no localitypreserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the Mpunctured sphere, localitypreserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local Fmoves and the mapping class group. This is joint work with Oliver Buerschaper, John Preskill, Robert Koenig, Fernando Pastawski and Sumit Sijher. 


