  We introduce a quantum control protocol that produces smooth, experimentally
implementable control sequences optimized to combat temporally correlated noise
for single qubit systems. The control ansatz is specifically chosen to be a
functional expansion of discrete prolate spheroidal sequences, a discrete time
basis known to be optimally concentrated in time and frequency, and quite
attractive when faced with experimental control hardware constraints. We
leverage the filter function formalism to transform the control problem into a
filter design problem, and show that the frequency response of a quantum system
can be carefully tailored to avoid the most relevant dynamical contributions of
noise processes. Using gradient ascent, we obtain optimized filter functions
and exploit them to elucidate important details about the relationship between
filter function design, control bandwidth, and noise characteristics. In
particular, we identify regimes of optimal noise suppression and in turn,
optimal control bandwidth directly proportional to the size of the frequency
bands where the noise power is large. In addition to providing guiding
principles for filter design, our approach enables the development of controls
that simultaneously yield robust noise filtering and high fidelity single qubit
logic operations in a wide variety of complex noise environments.
