  Counterdiabatic (CD) driving presents a way of generating adiabatic dynamics
at arbitrary pace, where excitations due to non-adiabaticity are exactly
compensated by adding an auxiliary driving term to the Hamiltonian. While this
CD term is theoretically known and given by the adiabatic gauge potential,
obtaining and implementing this potential in many-body systems is a formidable
task, requiring knowledge of the spectral properties of the instantaneous
Hamiltonians and control of highly nonlocal multibody interactions. We show how
an approximate gauge potential can be systematically built up as a series of
nested commutators, remaining well-defined in the thermodynamic limit.
Furthermore, the resulting CD driving protocols can be realized up to arbitrary
order without leaving the available control space using tools from
periodically-driven (Floquet) systems. This is illustrated on few- and
many-body quantum systems, where the resulting Floquet protocols significantly
suppress dissipation and provide a drastic increase in fidelity.
