Quantum computation with Turaev-Viro codes

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. 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.