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In this manuscript we address the problem of deriving \emph{analytic}expressions for calculating universal decoherence-induced errors in qubitsundergoing arbitrary, unitary, time-dependent quantum-control protocols. For aqubit undergoing unitary decoherence the evolution of a qubit state in thepresence of time-varying semiclassical fields may be treated geometrically. Weshow that the fidelity of an arbitrary control operation may then be expressedto arbitrary order in terms of experimentally relevant spectral characteristicsof the noise and the control over all Cartesian directions and accounting fornoise cross-correlations. We formulate \emph{control matrices} in the timedomain to capture the effects of piecewise-constant control, and convert themto generalized Fourier-domain filter functions. Such generalized filterfunctions may therefore be derived for complex temporally modulated controlprotocols, accounting for susceptibility to rotations of the qubit state vectorin three dimensions. Taken together, this framework provides a computationallyefficient means to calculate the effects of universal noise on arbitraryquantum control protocols without the need for time-consuming simulations ofBloch vector evolution. As a concrete example, we apply our method to treatingthe problem of dynamical decoupling incorporating realistic control pulses ofarbitrary duration or form, including the replacement of simple $\pi$-pulseswith complex dynamically corrected gates.

Arbitrary quantum control of qubits in the presence of universal noise