  Correlated, non-Markovian noise is present in many solid-state systems
employed as hosts for quantum information technologies, significantly
complicating the realistic theoretical description of these systems. In this
regime, the effects of noise on sequences of quantum gates cannot be described
by concatenating isolated quantum operations if the environmental correlation
times are on the scale of the typical gate durations. The filter function
formalism has been successful in characterizing the decay of coherence under
the influence of such classical, non-Markovian environments and here we show it
can be applied to describe unital evolution within the quantum operations
formalism. We find exact results for the quantum process and a simple
composition rule for a sequence of operations. This enables the detailed study
of effects of noise correlations on algorithms and periodically driven systems.
Moreover, we point out the method's suitability for numerical applications and
present the open-source Python software package filter_functions. Amongst other
things, it facilitates computing the noise-averaged transfer matrix
representation of a unital quantum operation in the presence of universal
classical noise for arbitrary control sequences. We apply the presented methods
to selected examples.
