  Quantum state processing is one of the main tools of quantum technologies.
While real systems are complicated and/or may be driven by non-ideal control
they may nevertheless exhibit simple dynamics approximately confined to a
low-energy Hilbert subspace. Adiabatic elimination is the simplest
approximation scheme allowing us to derive in certain cases an effective
Hamiltonian operating in a low-dimensional Hilbert subspace. However, these
approximations may present ambiguities and difficulties hindering a systematic
improvement of their accuracy in larger and larger systems. Here we use the
Magnus expansion as a systematic tool to derive ambiguity-free effective
Hamiltonians. We show that the validity of the approximations ultimately
leverages only on a properly done coarse-graining in time of the exact
dynamics. We validate the accuracy of the obtained effective Hamiltonians with
suitably tailored fidelities of quantum operations.
