We present a general transfer-function approach to noise filtering in open-loop Hamiltonian engineering protocols for open quantum systems. We show how to identify a computationally tractable set of fundamental filter functions, out of which arbitrary transfer filter functions may be assembled up to arbitrary high order in principle. Besides avoiding the infinite recursive hierarchy of filter functions that arises in general control scenarios, this fundamental filter-functions set suffices to characterize the error suppression capabilities of the control protocol in both the time and frequency domain. We prove that the resulting notion of filtering order reveals conceptually distinct, albeit complementary, features of the controlled dynamics as compared to the order of error cancellation, traditionally defined in the Magnus sense. Examples and implications are discussed.PACS numbers: 03.67. Pp, 03.65.Yz, 03.67.Lx, 07.05.Dz Hamiltonian engineering via open-loop quantum control provides a versatile and experimentally validated framework for manipulating the dynamics of a broad class of open quantum systems [1]. Applications range from dynamical decoupling (DD), composite pulse sequences and dynamically corrected quantum gates (DCGs), to noise spectroscopy and quantum simulation -see e.g. [2][3][4][5][6][7][8] for recent contributions. In this context, generalized transfer filter function (FF) techniques motivated by control engineering are providing an increasingly important tool for understanding the dynamical response of the target system in Fourier space and for quantitatively analyzing the control performance [9][10][11][12]. In particular, this formalism has proved remarkably successful in predicting operational fidelities for a variety of control settings in recent trapped-ion experiments [13], as long as noise is sufficiently weak for low-order approximations to be viable.From a control-theory standpoint, a filtering approach to dynamical error suppression is desirable for a variety of reasons. Besides providing an open-loop counterpart to the transfer-function perspective that is central to both classical and quantum feedback networks [14], FF techniques allow, when available, for a substantially more efficient analysis of the underlying noisy dynamics than direct simulation [15,16]. Furthermore, unlike traditional time-dependent perturbative approaches (such as the Magnus expansion [1,17]), a frequency-space picture may open up new possibilities for tailoring control synthesis and optimization to specific spectral features of the noise. The existing FF framework suffers, however, from severe limitations. Even if formal expressions for gate fidelity may be given based on an infinite recursive hierarchy of generalized FFs [12], higher-order FFs become rapidly intractable. Thus, explicit calculations have largely focused thus far on single-qubit controlled dynamics in the presence of classical noise, by truncating this recursion to the lowest order and additionally exploiting Gaussian noise statistics. As...
