Abstract
We demonstrate how to do many computations for doubled topological phases with defects. These defects may be 1-dimensional domain walls or 0-dimensional point defects.Using Vec(S3) as a guiding example, we demonstrate how domain wall fusion and associators can be computed using generalized tube algebra techniques. These domain walls can be both between distinct or identical phases. Additionally, we show how to compute all possible point defects, and the fusion and associator data of these. Worked examples, tabulated data and Mathematica code are provided.
Highlights
Topological phases are of great interest in quantum information theory
Many of the simplest topological codes only allow limited gate sets to be implemented in a fault tolerant manner
We focus on the example Vec(S3), both to illustrate the methods, and to make more complete data available to researchers interested in universal quantum computation using ‘small’ topological phases
Summary
By leveraging their topological nature, quantum information can be stored and manipulated in a manner which is protected against any local noise[1,2,3,4] For such encodings to be useful in our attempts to build a quantum computer, it is necessary to find ways of implementing universal sets of logic gates. A hybrid approach has been proposed[5] This primarily relies on the (relatively) straightforward code for storage and manipulation, but included small ‘islands’ of a more powerful phase to leverage their computational power. Hybrid schemes such as that described above, require information to be exchanged between codes. The fusion categories that arise in this paper are Vec, Vec(Z/2Z), Vec(Z/3Z) and Vec(S3) (defined below)
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