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After quantum error correction was introduced in mid '90s, in subsequent years many of the classical analogues regarding the structure of code (such as singleton bound, GV bound etc) were obtained in the quantum setting as well. Of course, there was also the big break of stabilizer formalism, fault-tolerant computing and particular families of codes like Toric code and other surface codes.

What are the general directions and target questions in the area (like current status of decoding algorithms and codes with good parameters) and applications of it?

For example on the application side, one key area that we see nowadays is local codes and connections to qPCP or hamiltonian complexity. What are some (big or small) questions that we are trying to answer?

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  • $\begingroup$ I remember something from the paper *Perfect zero knowledge for quantum multiprover interactive proofs*(arxiv.org/abs/1905.11280). I never read the paper thoroughly, but they seem to use techniques from quantum fault-tolerance in the first part of their proof. So it sounds like an application of quantum fault-tolerance and perhaps there are connections to interactive proofs too. $\endgroup$
    – raycosine
    Jan 12, 2021 at 11:53

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One big open question is the existence of "good" quantum LDPC codes. These are stabilizer quantum error correcting codes with constant check weight, constant rate and linear distance. Most recently there was a series of breakthroughs on this front by Hastings, Haah and O'Donnell (arXiv:2009.03921), Breuckmann and Eberhardt (arXiv:2012.09271) and Panteleev and Kalachev (arXiv:2012.04068).

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