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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 controlthey may nevertheless exhibit simple dynamics approximately confined to alow-energy Hilbert subspace. Adiabatic elimination is the simplestapproximation scheme allowing us to derive in certain cases an effectiveHamiltonian operating in a low-dimensional Hilbert subspace. However, theseapproximations may present ambiguities and difficulties hindering a systematicimprovement of their accuracy in larger and larger systems. Here we use theMagnus expansion as a systematic tool to derive ambiguity-free effectiveHamiltonians. We show that the validity of the approximations ultimatelyleverages only on a properly done coarse-graining in time of the exactdynamics. We validate the accuracy of the obtained effective Hamiltonians withsuitably tailored fidelities of quantum operations.

Coarse-grained effective Hamiltonian via the Magnus Expansion for a three-level system