  Counter-diabatic driving (CD) is a technique in quantum control theory
designed to counteract nonadiabatic excitations and guide the system to follow
its instantaneous energy eigenstates, and hence has applications in state
preparation, quantum annealing, and quantum thermodynamics. However, in many
practical situations, the effect of the environment cannot be neglected, and
the performance of the CD is expected to degrade. To arrive at universal bounds
on the resulting error of CD in this situation we consider a driven spin-boson
model as a prototypical setup. The inequalities we obtain, in terms of either
the Bures angle or the fidelity, allow us to estimate the maximum error solely
characterized by the parameters of the system and the bath. By utilizing the
analytical form of the upper bound, we demonstrate that the error can be
systematically reduced through optimization of the external driving protocol of
the system. We also show that if we allow a time-dependent system-bath coupling
angle, the obtained bound can be saturated and realizes unit fidelity.
