  Over the past decade, a number of quantum processes have been proposed which
are logically consistent, yet feature a cyclic causal structure. However, there
is no general formal method to construct a process with an exotic causal
structure in a way that ensures, and makes clear why, it is consistent. Here we
provide such a method, given by an extended circuit formalism. This only
requires directed graphs endowed with Boolean matrices, which encode basic
constraints on operations. Our framework (a) defines a set of elementary rules
for checking the validity of any such graph, (b) provides a way of constructing
consistent processes as a circuit from valid graphs, and (c) yields an
intuitive interpretation of the causal relations within a process and an
explanation of why they do not lead to inconsistencies. We display how several
standard examples of exotic processes, including ones that violate causal
inequalities, are among the class of processes that can be generated in this
way; we conjecture that this class in fact includes all unitarily extendible
processes.
