TL;DR
This paper introduces homological stabilizer codes, a class of quantum error-correcting codes defined by graphs, unifying and characterizing known codes like toric and color codes through topological graph theory.
Contribution
It formalizes homological stabilizer codes, relates them to existing codes, and classifies codes without local logical operators based on graph properties.
Findings
All toric codes are equivalent to homological stabilizer codes on 4-valent graphs.
Topological color codes and toric codes correspond to distinct graph classes.
Codes without local logical operators are equivalent to either toric or color codes.
Abstract
In this paper we define homological stabilizer codes which encompass codes such as Kitaev's toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of topological graph theory to determine the graphs which are suitable as homological stabilizer codes. We then show that all toric codes are equivalent to homological stabilizer codes on 4-valent graphs. We show that the topological color codes and toric codes correspond to two distinct classes of graphs. We define the notion of label set equivalencies and show that under a small set of constraints the only homological stabilizer codes without local logical operators are equivalent to Kitaev's toric code or to the topological color codes.
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Coding theory and cryptography · Graph theory and applications