Homological stabilizer codes

doi:10.1016/j.aop.2012.11.007

DOE OSTI · 22157076

Homological stabilizer codes

Abstract

In this paper we define homological stabilizer codes on qubits 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. - Highlights: Black-Right-Pointing-Pointer We show that Kitaev's toric codes are equivalent to homological stabilizer codes on 4-valent graphs. Black-Right-Pointing-Pointer We show that toric codes and color codes correspond to homological stabilizer codes on distinct graphs. Black-Right-Pointing-Pointer We find and classify all 2D homological stabilizer codes. Black-Right-Pointing-Pointer We find optimal codes among the homological stabilizer codes.

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71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS CORRECTIONS DIAGRAMS ERRORS GRAPH THEORY LIMITING VALUES QUANTUM CRYPTOGRAPHY QUANTUM MECHANICS QUANTUM STATES QUBITS TOPOLOGY

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2013-03-15. Homological stabilizer codes. https://doi.org/10.1016/j.aop.2012.11.007

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