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One of the main methods for protecting quantum information againstdecoherence is to encode information in the ground subspace (or the low energysector) of a Hamiltonian with a large energy gap which penalizes errors fromenvironment. The protecting Hamiltonian is chosen such that its degenerateground subspace is an error detecting code for the errors caused by theinteraction with the environment. We consider environments with arbitrarynumber of local sites, e.g. spins, whose interactions among themselves arelocal and bounded. Then, assuming the system is interacting with a finitenumber of sites in the environment, we prove that, up to second order withrespect to the coupling constant, decoherence and relaxation are suppressed bya factor which grows exponentially fast with the ratio of energy penalty to thenorm of local interactions in the environment. The state may, however, stillevolve unitarily inside the code subspace due to the Lamb shift effect. In thecontext of adiabatic quantum computation, this means that the evolution insidethe code subspace is effectively governed by a renormalized Hamiltonian. Theresult is derived from first principles, without use of master equations ortheir assumptions, and holds even in the infinite temperature limit. We alsoprove that unbounded or non-local interactions in the environment at sites farfrom the system do not considerably modify the exponential suppression. Ourmain technical tool is a new bound on the decay of power spectral density athigh frequencies for local observables and many-body Hamiltonians with boundedand local interactions in a neighborhood around the support of the observable.

Exponential suppression of decoherence and relaxation of quantum systems using energy penalty