Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices

TL;DR

This paper predicts the accuracy thresholds of topological color codes on hexagonal and square-octagonal lattices using mappings to spin glass systems, providing insights into their error correction capabilities.

Contribution

It introduces a novel analysis of topological color codes on different lattices via duality and gauge symmetry, estimating their error thresholds.

Findings

Threshold for hexagonal lattice: approximately 0.1096

Threshold for square-octagonal lattice: approximately 0.1092

Both thresholds are slightly below the quantum Gilbert-Varshamov bound

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-p_c = 0.1096-8 $ for the hexagonal lattice and $1-p_c = 0.1092-3$ 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 $1-p_c = 0.110028$ for the quantum Gilbert-Varshamov bound with a zero encoding rate.