  Molecular nanomagnets may enable the implementation of quantum
error-correction codes exploiting the many spin levels naturally embedded
within a single molecule, a potentially crucial step towards scalable quantum
processors. To fully realize the potential of this approach, a microscopic
understanding of the errors corrupting quantum information encoded in a
molecular qudit is essential, together with the development of tailor-made
quantum error correction strategies. We address these central points by first
studying dephasing effects on the molecular spin qudit produced by the
interaction with surrounding nuclear spins, which are the dominant source of
qudit errors at low temperatures, through a complete many-body description of
the nuclear spin bath. Numerical quantum error correction codes are then
constructed, by means of a systematic optimization procedure based on
simulations of the coupled system-bath dynamics, that provide a striking
enhancement of the coherence time of the molecular computational unit.
