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This paper claims that TQFTs are a natural language for quantum information theory and arbitrary quantum measurements, that quantum error correcting codes "are effectively spacetimes", and that this applies to topological M-theory.

I've just had a look at section 5, "Topological M-theory as a QECC". It contains expositions of 3d gravity as a Chern-Simons gauge theory (5.2), 4d gravity according to Plebanski (5.3), and topological M-theory (5.4). There is absolutely nothing in section 5 about quantum error-correcting codes, and almost nothing about TQFTs.

In 5.3, Plebanski's approach to 4d gravity is described as occurring "within the framework of topological quantum field theory", but Plebanski's paper (their reference 111) was published in 1975, 13 years before the first notion of TQFT was formulated. It's actually about obtaining complex-valued metrics that solve the Einstein equations (part of the interest is that real-valued solutions can sometimes be obtained as sections of complex-valued solutions).

In 5.1, while introducing topological M-theory, they blithely remark that the S-duality of topological M-theory "shed[s] light on... self-dual loop quantum gravity in 4-dimensions". No citation is given, and I have not found a paper in the bibliography which might be the source of this claim. However, one of the coauthors is also author of https://arxiv.org/abs/1707.05347, where such a connection is asserted. Like all papers asserting a relationship between loop quantum gravity and some part of string theory, this claim should (in my opinion) be looked upon with extreme skepticism.

@mitchellporter you can always email the authors of the paper with your queries.

I am not sure how much responsive they will be, though.

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