  Many qubit implementations are afflicted by correlated noise not captured by
standard theoretical tools that are based on Markov approximations. While
independent gate operations are a key concept for quantum computing, it is
actually not possible to fully describe noisy gates locally in time if noise is
correlated on times longer than their duration. To address this issue, we
develop a method based on the filter function formalism to perturbatively
compute quantum processes in the presence of correlated classical noise. We
derive a composition rule for the filter function of a sequence of gates in
terms of those of the individual gates. The joint filter function allows to
efficiently compute the quantum process of the whole sequence. Moreover, we
show that correlation terms arise which capture the effects of the
concatenation and thus yield insight into the effect of noise correlations on
gate sequences. Our generalization of the filter function formalism enables
both qualitative and quantitative studies of algorithms and state-of-the-art
tools widely used for the experimental verification of gate fidelities like
randomized benchmarking, even in the presence of noise correlations.
