Title:

Completeness and the ZXcalculus

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 ZXcalculus, 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 errorcorrecting codes or measurementbased quantum computation. Furthermore, I show that the ZXcalculus is complete for the singlequbit Clifford+T group, which is approximately universal: any singlequbit unitary can be approximated to arbitrary accuracy using only Clifford gates and the Tgate. In experimental realisations of quantum computers, operations have to be approximated using some such finite gate set. Therefore this result implies that a wide range of realistic scenarios in quantum computation can be analysed graphically without loss of deductive power. 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. The toy theory for the simplest possible underlying system closely resembles stabilizer quantum mechanics, which is nonlocal; it thus offers insights into the similarities and differences between classical and quantum theories. I develop a graphical calculus similar to the ZXcalculus 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.
