Citation: 'A Categorical Overview of Quantum Error Correction', Adam Marks, draft, 2025
Web: https://reversible.io/ (It hasn't been put on ArXiV yet)
Tags: Mathematical, Non-local-codes, Monoidal-categories
In this paper, the author performs two main technical tasks. The author describes known constructions of quantum error correcting codes (specifically CSS codes, stabilizer codes, surface codes, and Tanner codes) as functors. Concretely, this means the following. Instead of only assigning input data (such as a chain complex) to a quantum error correcting code, the author must also take transformations on input data and coherently map them into transformations on the error correcting code. In some cases the author is fully successful, and in other cases they are only successful in mapping though the isomorphisms on input data. The author introduced a new sort of algebraic geometry called "meta" algebraic geometry and uses this to give a functorial description of the error correcting property of an error correcting code.
The author has several technical results, but this paper still feels like a work in progress. I was not able to locate any of the functorial constructions of error correcting codes that felt non-trivial, in the sense that it all felt like re-packaging of known constructions. The most non-trivial part of the work seems to be the introduction of a new "meta" algebraic geometry and its use for describing the error correction property of error correcting codes. However, this feels quite unsatisfying to me. I don't want to work with a new algebraic geometry if it isn't strictly necessary. The point of the "meta" algebraic geometry is to make complex conjugation a polynomial. There are more conventional alternate ways of doing this - for instance, the author could have embedded n complex variables into 2n real variables, which turns complex conjugation into a valid real polynomial. I have trouble feeling like this new perspective helps organize my thinking because it requires me to introduce a new structure I am suspicious of. The author leaves a large amount of the potential applications of the formalism to "future work" in the conclusions section, emphasizing the feeling that this is a work in progress.
This isn't the framing the author takes, but the perspective of functoriality as logical operations could be of broader interest, since constructing logical operations is such an important topic. One of the important sources of logical gates in topological codes (such as the surface code, included in the scope of the article) is the induced action of the mapping class group of a surface on the associated code. That is, these logical gates on the code come from pushing forward the logical operations on the structure used to create the code. This is exactly the sort of the thing the author is talking about. Coming up with new ways to make code constructions functorial is exactly the problem of translating operations on underlying structures used to make codes into logical operations.