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IEEE International Symposium on Information Theory (ISIT)

New classes of topological quantum codes derived from embeddings of graphs on compact surfaces

Clarice Dias de Albuquerque, Reginaldo Palazzo, Eduardo Brandani da Silva

Published Jun 1, 2010• 0 citations
Publisher

Abstract

In this paper we present classes of topological quantum codes (TQC) derived from the embedding of complete bipartite graphs on the corresponding compact surfaces. Every code achieves minimum distance 3 and its encoding rate is such that k/n → 1 as n → ∞.
DOI
10.1109/isit.2010.5513645

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In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus g≥2g\ge 2g≥2. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that kn→1\frac{k}{n} \rightarrow 1nk​→1 as n→∞n \rightarrow \inftyn→∞, except for the one case where kn→13\frac{k}{n} \rightarrow \frac{1}{3}nk​→31​ as n→∞n \rightarrow \inftyn→∞.
In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g≥2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2–5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s≡1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3–5.
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