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Quantum double aspects of surface code models
- Publication Year :
- 2021
-
Abstract
- We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry, where $G$ is a finite group. We provide projection operators for its quasiparticles content as irreducible representations of $D(G)$ and combine this with $D(G)$-bimodule properties of open ribbon excitation spaces $L(s_0,s_1)$ to show how open ribbons can be used to teleport information between their endpoints $s_0,s_1$. We give a self-contained account that builds on earlier work but emphasises applications to quantum computing as surface code theory, including gates on $D(S_3)$. We show how the theory reduces to a simpler theory for toric codes in the case of $D( \Bbb Z_n)\cong \Bbb C\Bbb Z_n^2$, including toric ribbon operators and their braiding. In the other direction, we show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$, including site actions of $D(H)$ and partial results on ribbon equivariance even when the Hopf algebra is not semisimple.<br />Comment: 54 pages, many figures both pdf and tkz
- Subjects :
- Quantum Physics
Mathematical Physics
Mathematics - Quantum Algebra
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2107.04411
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1063/5.0063768