  The ZX-calculus is a convenient formalism for expressing and reasoning about
quantum circuits at a low level, whereas the recently-proposed ZH-calculus
yields convenient expressions of mid-level quantum gates such as Toffoli and
CCZ. In this paper, we will show that the two calculi are linked by Fourier
transform. In particular, we will derive new Fourier expansion rules using the
ZH-calculus, and show that we can straightforwardly pass between ZH- and
ZX-diagrams using them. Furthermore, we demonstrate that the graphical Fourier
expansion of a ZH normal-form corresponds to the standard Fourier transform of
a semi-Boolean function. As an illustration of the calculational power of this
technique, we then show that several tricks for reducing the T-gate cost of
Toffoli circuits, which include for instance quantum adders, can be derived
using graphical Fourier theory and straightforwardly generalized to more
qubits.
