title: Quantum error suppression with commuting Hamiltonians: two local is too local.
abstract: We consider error suppression schemes in which quantum information is encoded into the ground subspace of a Hamiltonian comprising a sum of commuting terms. Since such Hamiltonians are gapped, they are considered natural candidates for protection of quantum information and topological or adiabatic quantum computation. However, we prove that they cannot be used to this end in the two-local case. By making the favorable assumption that the gap is infinite, we show that single-site perturbations can generate a degeneracy splitting in the ground subspace of this type of Hamiltonian which is of the same order as the magnitude of the perturbation, and is independent of the number of interacting sites and their Hilbert space dimensions, just as in the absence of the protecting Hamiltonian. This splitting results in decoherence of the ground subspace, and we demonstrate that for natural noise models the coherence time is proportional to the inverse of the degeneracy splitting. Our proof involves a new version of the no-hiding theorem which shows that quantum information cannot be approximately hidden in the correlations between two quantum systems. The main reason that two-local commuting Hamiltonians cannot be used for quantum error suppression is that their ground subspaces have only short-range (two-body) entanglement.
authors.affiliation: Center for Quantum Information Science and and Technology, University of Southern California, Los Angeles, California 90089, USA and Department of Physics, University of Southern California, Los Angeles, California 90089, USA.
authors.forename: Iman
authors.initials: I
authors.lastname: Marvian
authors.affiliation: Center for Quantum Information Science and and Technology, University of Southern California, Los Angeles, California 90089, USA and Department of Physics, University of Southern California, Los Angeles, California 90089, USA and Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA and Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA.
authors.forename: Daniel A
authors.initials: DA
authors.lastname: Lidar