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We consider error suppression schemes in which quantum information is encodedinto the ground subspace of a Hamiltonian comprising a sum of commuting terms.Since such Hamiltonians are gapped they are considered natural candidates forprotection of quantum information and topological or adiabatic quantumcomputation. However, we prove that they cannot be used to this end in the2-local case. By making the favorable assumption that the gap is infinite weshow that single-site perturbations can generate a degeneracy splitting in theground subspace of this type of Hamiltonians which is of the same order as themagnitude of the perturbation, and is independent of the number of interactingsites and their Hilbert space dimensions, just as in the absence of theprotecting Hamiltonian. This splitting results in decoherence of the groundsubspace, and we demonstrate that for natural noise models the coherence timeis proportional to the inverse of the degeneracy splitting. Our proof involvesa new version of the no-hiding theorem which shows that quantum informationcannot be approximately hidden in the correlations between two quantum systems,and should be of independent interest. The main reason that 2-local commutingHamiltonians cannot be used for quantum error suppression is that their groundsubspaces have only short-range (two-body) entanglement.

Quantum error suppression with commuting Hamiltonians: Two-local is too local