Hamiltonian engineering via open-loop quantum control provides a versatile and experimentally validated framework for precisely manipulating a broad class of non-Markovian dynamical evolutions of interest, with applications ranging from dynamical decoupling and dynamically corrected quantum gates to noise spectroscopy and quantum simulation. In this context, transfer-function techniques directly motivated by control engineering have proved invaluable for obtaining a transparent picture of the controlled dynamics in the frequency domain and for quantitatively analyzing control performance. In this talk, I will show how to construct a general filter-function approach, which overcomes the limitations of the existing formalism. The key insight is to identify a set of "fundamental filter functions", whose knowledge suffices to construct arbitrary filter functions in principle and to determine the minimum "filtering order" that a given control protocol can guar antee. Implications for dynamical control in multi-qubit systems and/or in the presence of non-Gaussian noise will be discussed.
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