  The ZX-calculus is a powerful diagrammatic language for quantum mechanics and
quantum information processing. We prove that its \pi/4-fragment is not
complete, in other words the ZX-calculus is not complete for the so called
"Clifford+T quantum mechanics". The completeness of this fragment was one of
the main open problems in categorical quantum mechanics, a programme initiated
by Abramsky and Coecke. The ZX-calculus was known to be incomplete for quantum
mechanics. On the other hand, its \pi/2-fragment is known to be complete, i.e.
the ZX-calculus is complete for the so called "stabilizer quantum mechanics".
Deciding whether its \pi/4-fragment is complete is a crucial step in the
development of the ZX-calculus since this fragment is approximately universal
for quantum mechanics, contrary to the \pi/2-fragment. To establish our
incompleteness result, we consider a fairly simple property of quantum states
called supplementarity. We show that supplementarity can be derived in the
ZX-calculus if and only if the angles involved in this equation are multiples
of \pi/2. In particular, the impossibility to derive supplementarity for \pi/4
implies the incompleteness of the ZX-calculus for Clifford+T quantum mechanics.
As a consequence, we propose to add the supplementarity to the set of rules of
the ZX-calculus. We also show that if a ZX-diagram involves antiphase twins,
they can be merged when the ZX-calculus is augmented with the supplementarity
rule. Merging antiphase twins makes diagrammatic reasoning much easier and
provides a purely graphical meaning to the supplementarity rule.
