Quantum double aspects of surface code models

TL;DR

This paper explores the quantum double models of surface codes, detailing their quasiparticle structure, ribbon operators, and applications to quantum computing, including generalizations to Hopf algebra-based models.

Contribution

It provides a comprehensive framework for understanding surface code models using quantum doubles, with explicit constructions and applications to quantum gates and generalizations.

Findings

Projection operators for quasiparticles as irreducible representations of D(G)

Ribbon operators enable teleportation of information between endpoints

Reduction to simpler toric code models for D(Z_n)

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.