Quantum computation with Turaev-Viro codes

          @misc{ pirsa_PIRSA:17080010,
            doi = {10.48660/17080010},
            url = {https://pirsa.org/17080010},
            author = {Koenig, Robert},
            keywords = {Quantum Foundations, Quantum Information},
            language = {en},
            title = {Quantum computation with Turaev-Viro codes},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {aug},
            note = {PIRSA:17080010 see, \url{https://pirsa.org}}
          }
          

Abstract

The Turaev-Viro invariant for a closed 3-manifold is defined as the contraction of a certain tensor network. The tensors correspond to tetrahedra in a triangulation of the manifold, with values determined by a fixed spherical category. For a manifold with boundary, the tensor network has free indices that can be associated to qudits, and its contraction gives the coefficients of a quantum error-correcting code. The code has local stabilizers determined by Levin and Wen. By studying braid group representations acting on equivalence classes of colored ribbon graphs embedded in a punctured sphere, we identify the anyons, and give a simple recipe for mapping fusion basis states of the doubled category to ribbon graphs. Combined with known universality results for anyonic systems, this provides a large family of schemes for quantum computation based on local deformations of stabilizer codes. These schemes may serve as a starting point for developing fault-tolerance schemes using continuous stabilizer measurements and active error-correction. This is joint work with Greg Kuperberg and Ben Reichardt.