  We introduce the fermionic ZW calculus, a string-diagrammatic language for
fermionic quantum computing (FQC). After defining a fermionic circuit model, we
present the basic components of the calculus, together with their
interpretation, and show how the main physical gates of interest in FQC can be
represented in our language. We then list our axioms, and derive some
additional equations. We prove that the axioms provide a complete equational
axiomatisation of the monoidal category whose objects are systems of finitely
many local fermionic modes (LFMs), with maps that preserve or reverse the
parity of states, and the tensor product as monoidal product. We achieve this
through a procedure that rewrites any diagram in a normal form. As an example,
we show how the statistics of a fermionic Mach-Zehnder interferometer can be
calculated in the diagrammatic language. We conclude by giving a diagrammatic
treatment of the dual-rail encoding, a standard method in optical quantum
computing used to perform universal quantum computation.
