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Recently, it has become apparent that the thermal stability of topologicallyordered systems at finite temperature, as discussed in condensed matterphysics, can be studied by addressing the feasibility of self-correctingquantum memory, as discussed in quantum information science. Here, with thiscorrespondence in mind, we propose a model of quantum codes that may cover alarge class of physically realizable quantum memory. The model is supported bya certain class of gapped spin Hamiltonians, called stabilizer Hamiltonians,with translation symmetries and a small number of ground states that does notgrow with the system size. We show that the model does not work asself-correcting quantum memory due to a certain topological constraint ongeometric shapes of its logical operators. This quantum coding theoreticalresult implies that systems covered or approximated by the model cannot havethermally stable topological order, meaning that no system can be stableagainst both thermal fluctuations and local perturbations simultaneously in twoand three spatial dimensions.

Feasibility of self-correcting quantum memory and thermal stability of topological order