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## "A hierarchy of anyon models realised by twists in stacked surface codes", Thomas Scruby, Dan Browne, 2020

*Reviewed September 4, 2023*

*Citation:* Scruby, T. R., and Dan E. Browne. "A hierarchy of anyon models realised by twists in stacked surface codes." Quantum 4 (2020): 251.

*Web:* https://quantum-journal.org/papers/q-2020-04-06-251/

*Tags:* Abelian-anyons, Hadamard-matrices

This paper brings the world of domain boundaries and twists defects to the world of mortals. That is,
it gives everything in terms of explicit formulas and does away with the category-theoretic formalism.
Categories can not be removed in all cases; they only do it for a subset of instances.

Additionally, there is a connection shown between F-matricies and Hadamard matrices shown. There is a long standing
conjecture, stating that Hadamard matrices exist in every dimension which is a multiple of 4.
Hence, the resolution of the Hadamard conjecture may reduce to the construction of topological phases.
It's not surprising that Hadamard matrices appear in this context, however. One very powerful
construction of type II1 subfactors (Jones' favorite, according to a lecture I watched)
comes from Hadmard matrices. The existence of Hadamard matricies thus implies the existence of exotic subfactors.
Subfactors and tensor categories are intimately linked.

What's different about this result from Jones' is that not every subfactor gives a Hadamard matrix.
However, in this case every good tensor category has F-matricies, which all give Hadamard matricies.
Hence, this really might be a viable proof method...