Extremal Optimization for estimation of the error threshold in topological subsystem codes at $\mathbf{T=0}$

Quantum decoherence is a problem that arises in implementations of quantum computing proposals. Topological subsystem codes (\emph{TSC}) have been suggested as a way to overcome decoherence. These offer a higher optimal error tolerance when compared to typical error-correcting algorithms. A TSC has been translated into a planar Ising spin-glass with constrained bimodal three-spin couplings. This spin-glass has been considered at finite temperature to determine the phase boundary between the unstable phase and the stable phase, where error recovery is possible.\footnote{R. S. Andrist et al., \emph{Optimal error correction in topological subsystem codes}, Phys. Rev. A., \textbf{85}, 050302(R) (2012)} We approach the study of the error threshold problem by exploring ground states of this spin-glass with the Extremal Optimization algorithm (\emph{EO}).\footnote{S. Boettcher et al., \emph{Optimization with extremal dynamics}, Phys. Rev. Lett., \textbf{86}, 5211 (2001)} EO has proven to be a effective heuristic to explore ground state configurations of glassy spin-systems.\footnote{S. Boettcher, \emph{Stiffness of the Edwards-Anderson model in all dimensions}, Phys. Rev. Lett., \textbf{95}, 197205 (2005)}