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Pauli Center for Theoretical Studies

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Quantum Science and Technology

CQT Singapore

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Sandia National Laboratories

Institute for Quantum Computing

id Quantique

Quantum computation with Turaev-Viro codes 

Robert Koenig, Caltech

joint work with Greg Kuperberg, and Ben Reichardt

The Turaev-Viro invariant for a closed three-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 Z2 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 (Levin-Wen string nets). 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.