  Given the rising popularity of quantum machine learning (QML), it is
important to develop techniques that effectively simplify commonly adopted
families of parameterised quantum circuits (commonly known as ans\"{a}tze).
This thesis pioneers the use of diagrammatic techniques to reason with QML
ans\"{a}tze. We take commonly used QML ans\"{a}tze and convert them to
diagrammatic form and give a full description of how these gates commute,
making the circuits much easier to analyse and simplify. Furthermore, we
leverage a combinatorial description of the interaction between CNOTs and phase
gadgets to analyse a periodicity phenomenon in layered ans\"{a}tze and also to
simplify a class of circuits commonly used in QML.
