Quantum computation with Turaev-Viro codes
Robert Koenig, Greg Kuperberg, Ben W. Reichardt

TL;DR
This paper introduces a framework for quantum computation using Turaev-Viro codes derived from tensor networks associated with 3-manifold invariants, enabling topological quantum computing with both abelian and non-abelian anyons.
Contribution
It presents a novel construction of quantum error-correcting codes from Turaev-Viro invariants, linking tensor networks, topological invariants, and anyon models for quantum computation.
Findings
Realizes non-abelian anyon models like Fibonacci over a lattice.
Provides methods for efficient state preparation, braiding, and measurement.
Establishes a connection between tensor network invariants and topological quantum 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…
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