  Instabilities due to extrinsic interference are routinely faced in systems
engineering, and a common solution is to rely on a broad class of
$\textit{filtering}$ techniques in order to afford stability to intrinsically
unstable systems. For instance, electronic systems are frequently designed to
incorporate electrical filters composed of, $\textit{e.g.}$ RLC components, in
order to suppress the effects of out-of-band fluctuations that interfere with
desired performance. Quantum coherent systems are now moving to a level of
complexity where challenges associated with realistic time-dependent noise are
coming to the fore. Unfortunately, standard control solutions involving
feedback are generally impossible due to the strictures of quantum mechanics,
and existing error-suppressing gate constructions generally rely on unphysical
bang-bang controls or quasi-static error models that do not reflect realistic
laboratory environments. In this work we use the theory of quantum control
engineering and experiments with trapped $^{171}$Yb$^{+}$ ions to demonstrate
the construction of novel $\textit{noise filters}$ which are specifically
designed to mitigate the effect of realistic time-dependent fluctuations on
qubits \emph{during useful operations}. Starting with desired filter
characteristics and the Walsh basis functions, we use a combination of analytic
design rules and numeric search to construct time-domain noise filters tailored
to a desired state transformation. Our results validate the generalized
filter-transfer function framework for arbitrary quantum control operations,
and demonstrate that it can be leveraged as an effective and efficient tool for
developing novel robust control protocols.
