  We determine the minimum energy required to control the evolution of any
mesoscopic quantum system in the presence of arbitrary Markovian noise
processes. This result provides the mesoscopic equivalent of the fundamental
cost of refrigeration, sets the minimum power consumption of mesoscopic devices
that operate out of equilibrium, and allows one to calculate the efficiency of
any control protocol, whether it be open-loop or feedback control. As examples
we calculate the energy cost of maintaining a qubit in the ground state, the
efficiency of resolved-sideband cooling of nano-mechanical resonators, and
discuss the energy cost of quantum information processing.
