  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. For example, applied to the genus-one handlebody using the Z_2
category, this construction yields the well-known toric code.
  For other categories, such as the Fibonacci category, the construction
realizes a non-abelian anyon model over a discrete lattice. 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. We explain how suitable initial states can be prepared efficiently, how
to implement braids, by successively changing the triangulation using a fixed
five-qudit local unitary gate, and how to measure the topological charge.
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.
