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## "The boundaries and twist defects of the color code and their applications to topological quantum computation", Markus Kesselring et al., 2018

*Reviewed January 14, 2024*

*Citation:* Kesselring, Markus S., et al. "The boundaries and twist defects of the color code and their applications to topological quantum computation." Quantum 2 (2018): 101.

*Web:* https://quantum-journal.org/papers/q-2018-10-19-101/

*Tags:* Abelian-anyons, Ising-computer, Defects/boundaries, Error-correcting-codes

This papers gives an amazing deep-dive into twist defects in the color code.
The classification of possible symmetries of the color code is very simple:
they form a group of order 72. It is then shown that all of these symmetries
can be realized physically as two dimensional nearest neighbor interactions
on the color code. This serves as the foundational work for any future
work which desires to exploit color code twist defects.

It is good to remember why we are studying the color code instead of, say, the toric code.
The toric code has two symmetries, only one of which is non-trivial. The color code has 72.
Symmetries correspond to twist defects, and so the highly symmetric nature of the color code
gives it a huge amount more twist defects. One philosophical way to understand this is as follows.
Not only is the color code isomorphic to bilayer toric code, but it is also isomorphic to bilayer 3-fermion.
All of the particles in the 3-fermion model are identical, so it has an S3 symmetry group. Two layers
of 3-fermion give an (S3 x S3) symmetry group, and the layer exchange symmetry adds a semidirect product with Z2.

The fact that bilayer 3-fermion and bilayer toric code can be isomorphic is a bit surprising.
For groups, decomposition into product factors is unique by the Krull-Schmidt theorem.
However, MTCs are not groups so counterintuitive properties are abound. The fact that
color code is isomorphic to bilayer 3-fermion is a nice construction, which
was not present in the literature before this and was pointed out to the authors by Zhenghan Wang.

As an application of their methods, the authors construct all sorts of record-breaking
codes, achieving fault-tolerant quantum computation with less physical qubits than before.
These codes can be seen as improvements of the ones in

> Yoder, Theodore J., and Isaac H. Kim. "The surface code with a twist." Quantum 1 (2017): 2.