computations and information can in principle be protected from decoherence. However, the need for such an active control mechanism poses a major challenge for any physical implementation. It is therefore of greatest interest to look for passively protected systems which are intrinsically stable against the destructive influence of a thermal environment. For this reason, the idea to encode quantum information in a topologically ordered ground state $|\Psi_0\rangle$ of a suitable Hamiltonian has attracted a lot of interest [@Kitaev2003; @Dennis2002; @Kitaev2006; @Bacon2006; @Trebst2007; @Tupitsyn2008; @Vidal2009a; @Vidal2009b; @Nussinov2008; @Alicki2009; @Iblisdir2009; @Iblisdir2010; @Bravyi2009; @Kay2008; @Pastawski2009; @Pastawski2009b; @Hamma2009; @Chesi2009a].

Important candidates among such topological models are stabilizer Hamiltonians [@Nielsen2000; @Gottesman1997], which are given by a sum of mutually commuting many-body Pauli operators. The advantage of such Hamiltonians is that the full energy spectrum is known and error correction schemes are readily derived [@Nielsen2000; @Gottesman1997]. A prototypical example of such models is the toric code proposed in Ref. [@Kitaev2003], for which the stability against Hamiltonian perturbations [@Trebst2007; @Tupitsyn2008; @Vidal2009a; @Vidal2009b; @Pastawski2009b] and thermal fluctuations [@Dennis2002; @Nussinov2008; @Alicki2009; @Iblisdir2009; @Iblisdir2010] was studied extensively. However, recent results [@Bravyi2009; @Kay2008] show that in one and two spatial dimensions no stabilizer Hamiltonian with finite-range interactions (including the toric code 