The ability to protect quantum information from the effect of noise is one of
the major goals of quantum information processing. In this article, we study
limitations on the asymptotic stability of quantum information stored in
passive N-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 prove that, regardless of the
protecting Hamiltonian, there exists a perturbed evolution that necessitates a
final error correcting step when the state of the memory is read. Such an error
correction step is shown to require a finite error threshold, the lack thereof
being exemplified by the 3D compass model. 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)).Comment: 17 pages and appendice