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We study the long-time evolution of a quantum memory coupled to a bosonicenvironment on which quantum error correction (QEC) is performed using thesurface code. The memory's evolution encompasses $N$ QEC cycles, each of themyielding a non-error syndrome. This assumption makes our analysis independentof the recovery process. We map the expression for the time evolution of thememory onto the partition function of an equivalent statistical mechanical spinsystem. In the superohmic dissipation case the long-time evolution of thememory has the same behavior as the time evolution for just one QEC cycle. Forthis case we find analytical expressions for the critical parameters of theorder-disorder phase transition of an equivalent spin system. These criticalparameters determine the threshold value for the system-environment couplingbelow which it is possible to preserve the memory's state.

Long-time efficacy of the surface code in the presence of a superohmic environment