General bound on the performance of counter-diabatic driving acting on dissipative spin systems

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 general 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.