  We argue that the quantum-theoretical structures studied in several recent
lines of research cannot be adequately described within the standard framework
of quantum circuits. This is in particular the case whenever the combination of
subsystems is described by a nontrivial blend of direct sums and tensor
products of Hilbert spaces. We therefore propose an extension to the framework
of quantum circuits, given by \textit{routed linear maps} and \textit{routed
quantum circuits}. We prove that this new framework allows for a consistent and
intuitive diagrammatic representation in terms of circuit diagrams, applicable
to both pure and mixed quantum theory, and exemplify its use in several
situations, including the superposition of quantum channels and the causal
decompositions of unitaries. We show that our framework encompasses the
`extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)],
which we derive as a special case, endowing them with a sound semantics.
