  The causal structure of a unitary transformation is the set of relations of
possible influence between any input subsystem and any output subsystem. We
study whether such causal structure can be understood in terms of compositional
structure of the unitary. Given a quantum circuit with no path from input
system $A$ to output system $B$, system $A$ cannot influence system $B$.
Conversely, given a unitary $U$ with a no-influence relation from input $A$ to
output $B$, it follows from [B. Schumacher and M. D. Westmoreland, Quantum
Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit
decomposition of $U$ with no path from $A$ to $B$. However, as we argue, there
are unitaries for which there does not exist a circuit decomposition that makes
all causal constraints evident simultaneously. To address this, we introduce a
new formalism of `extended circuit diagrams', which goes beyond what is
expressible with quantum circuits, with the core new feature being the ability
to represent direct sum structures in addition to sequential and tensor product
composition. A causally faithful extended circuit decomposition, representing a
unitary $U$, is then one for which there is a path from an input $A$ to an
output $B$ if and only if there actually is influence from $A$ to $B$ in $U$.
We derive causally faithful extended circuit decompositions for a large class
of unitaries, where in each case, the decomposition is implied by the unitary's
respective causal structure. We hypothesize that every finite-dimensional
unitary transformation has a causally faithful extended circuit decomposition.
