  Graphical languages offer intuitive and rigorous formalisms for quantum
physics. They can be used to simplify expressions, derive equalities, and do
computations. Yet in order to replace conventional formalisms, rigour alone is
not sufficient: the new formalisms also need to have equivalent deductive
power. This requirement is captured by the property of completeness, which
means that any equality that can be derived using some standard formalism can
also be derived graphically.
  In this thesis, I consider the ZX-calculus, a graphical language for pure
state qubit quantum mechanics. I show that it is complete for pure state
stabilizer quantum mechanics, so any problem within this fragment of quantum
theory can be fully analysed using graphical methods. This includes questions
of central importance in areas such as error-correcting codes or
measurement-based quantum computation. Furthermore, I show that the ZX-calculus
is complete for the single-qubit Clifford+T group, which is approximately
universal: any single-qubit unitary can be approximated to arbitrary accuracy
using only Clifford gates and the T-gate. [...] Lastly, I extend the use of
rigorous graphical languages outside quantum theory to Spekkens' toy theory, a
local hidden variable model that nevertheless exhibits some features commonly
associated with quantum mechanics. [...] I develop a graphical calculus similar
to the ZX-calculus that fully describes Spekkens' toy theory, and show that it
is complete. Hence, stabilizer quantum mechanics and Spekkens' toy theory can
be fully analysed and compared using graphical formalisms.
  Intuitive graphical languages can replace conventional formalisms for the
analysis of many questions in quantum computation and foundations without loss
of mathematical rigour or deductive power.
