  We introduce an object called a \emph{subspace graph} that formalizes the
technique of multidimensional quantum walks. Composing subspace graphs allows
one to seamlessly combine quantum and classical reasoning, keeping a classical
structure in mind, while abstracting quantum parts into subgraphs with simple
boundaries as needed. As an example, we show how to combine a \emph{switching
network} with arbitrary quantum subroutines, to compute a composed function. As
another application, we give a time-efficient implementation of quantum Divide
\& Conquer when the sub-problems are combined via a Boolean formula. We use
this to quadratically speed up Savitch's algorithm for directed
$st$-connectivity.
