  Adiabatic quantum computing (AQC) can be protected against thermal
excitations via an encoding into error detecting codes, supplemented with an
energy penalty formed from a sum of commuting Hamiltonian terms. Earlier work
showed that it is possible to suppress the initial thermally induced excitation
out of the encoded ground state, in the case of local Markovian environments,
by using an energy penalty strength that grows only logarithmically in the
system size, at a fixed temperature. The question of whether this result
applies beyond the initial time was left open. Here we answer this in the
affirmative. We show that thermal excitations out of the encoded ground state
can be suppressed at arbitrary times under the additional assumption that the
total evolution time is polynomial in the system size. Thus, computational
problems that can be solved efficiently using AQC in a closed system setting,
can still be solved efficiently subject to coupling to a thermal environment.
Our construction uses stabilizer subspace codes, which require at least
$4$-local interactions to achieve this result.
