We consider a ZX-calculus augmented with triangle nodes which is well-suited
to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We
precisely show the form of the matrices it represents, and we provide an
axiomatisation which makes the language complete for the Toffoli-Hadamard
quantum mechanics. We extend the language with arbitrary angles and show that
any true equation involving linear diagrams which constant angles are multiple
of Pi are derivable. We show that a single axiom is then necessary and
sufficient to make the language equivalent to the ZX-calculus which is known to
be complete for Clifford+T quantum mechanics. As a by-product, it leads to a
new and simple complete axiomatisation for Clifford+T quantum mechanics.Comment: In Proceedings QPL 2018, arXiv:1901.09476. Contains Appendi
