Citation: Qiu, Yang, and Zhenghan Wang. "Ground subspaces of topological phases of matter as error correcting codes." Annals of Physics 422 (2020): 168318.
Web: https://arxiv.org/abs/2004.11982
Tags: TQFT, Error-correcting-codes
This is one of the first papers showing that the ground states of very general topological phases form error correcting codes. The first paper of this sort was
> Cui, Shawn X., et al. "Kitaev's quantum double model as an error correcting code." Quantum 4 (2020): 331.
Which applied to all quantum doubles of finite groups using Kitaev's Hamiltonian. This paper shows that the ground states of doubled phases all form error correcting codes, using a TQFT perspective. Namely, the Levin-Wen/Turaev-Viro model. The connection between the Levin-Wen model and the Turaev-Viro model was established in
> Kirillov Jr, Alexander. "String-net model of Turaev-Viro invariants." arXiv preprint arXiv:1106.6033 (2011).
This paper uses a TQFT formalism similar to Kirillov's, so that they can apply his results. The authors also use their proof technique to show that the ground states of the Dijkgraaf-Witten model form an error correcting code.
This paper is a fantastic reference for how one makes the jump from TQFTs to gapped Hamiltonians/error correcting codes. Zhenghan makes a very clear analogy between the axioms of a TQFT and the axioms of an error correcting code. The discussion of gapped Hamiltonian schemas and their limitations is very good.
This paper is very honest about what is known and what is not. This makes it a good place to learn the state of the field circa 2020. One issue with this paper is that it is somewhat sloppily written and notation is inconsistent.
A nice quote: "in reality, not all celluations should be allowed as lattices for real materials are usually highly constrained by quantum chemistry". This makes the physical connection all the way down very clear.