-qubit systems. We consider the effect of small imperfections in the implementation of the protecting Hamiltonian in the form of perturbations or weak coupling to a ground state environment. We thus depart from the usual Markovian approximation for a thermal bath by conce ntrating on models for which part of the evolution can be calculated exactly. We prove that, regardless of the protecting Hamiltonian, there exists a perturbed evolution that necessitates a final error correcting step for the state of the memory to be read. Such an error correction step is shown to require a finite error threshold, the lack thereof being exemplified by the 3D XZ-compass model \cite{bacon-2006}. We go on to present explicit weak Hamiltonian perturbations which destroy the logical information stored in the 2D toric code in a time

O(\log(N))

DOI: 10.26421/qic10.7-8-3

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N

O(\log(N))

{\mathbb{Z}}_{2}

symmetry breaking in a Lindbladian is useful to protect a classical bit while intrinsic topological order protects a qubit.