## Abstract

Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the square-octagonal lattice by the use of mapping into the spin-glass systems. The analysis for the corresponding spin-glass systems consists of the duality, and the gauge symmetry, which has succeeded in deriving locations of special points, which are deeply related with the accuracy thresholds of topological error correcting codes. We predict that the accuracy thresholds for the topological color codes would be 1- pc =0.1096-8 for the hexagonal lattice and 1- pc =0.1092-3 for the square-octagonal lattice, where 1-p denotes the error probability on each qubit. Hence, both of them are expected to be slightly lower than the probability 1- pc =0.110028 for the quantum Gilbert-Varshamov bound with a zero encoding rate.

Original language | English |
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Article number | 011141 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 80 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Aug 6 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics