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"Topological quantum computation with gapped boundaries", Iris Cong, Meng Cheng, Zhenghan Wang, 2015

Reviewed August 21, 2023

Citation: Cong, Iris, Meng Cheng, and Zhenghan Wang. "Topological quantum computation with gapped boundaries." arXiv preprint arXiv:1609.02037 (2016).

Web: https://arxiv.org/abs/1609.02037

Tags:Defects/boundaries, Abelian-anyons, Modular-tensor-categories

This paper gives a proper treatment of the mathematics of gapped-boundary TQFTs, and their applications to topological quantum computing. A gapped boundary system is one in which not only does the bulk have a topological degeneracy, but the degeneracy extends to the boundary as well. That is, the Hamiltonian extends to having "boundary terms". There are multiple different ways of choosing these boundary terms, parameterized by Lagrangian algebras.

This is very important because it allows one to generalize the surface-code like method of storing information in holes (i.e. degeneracies of manifolds with boundaries) to non-abelian anyons, which is not a-priori obvious. A nice discussion of this extension with explicit Hamiltonians and circuits is given.

This paper is very long (117 pages) and hence contains both a lot of general theory as well as specific examples. Namely, special case is given to the case of G=Z3 because this is a phase which appears in real fractional quantum Hall experiments.