  The field of quantum machine learning (QML) explores how quantum computers
can be used to more efficiently solve machine learning problems. As an
application of hybrid quantum-classical algorithms, it promises a potential
quantum advantages in the near term. In this thesis, we use the ZXW-calculus to
diagrammatically analyse two key problems that QML applications face.
  First, we discuss algorithms to compute gradients on quantum hardware that
are needed to perform gradient-based optimisation for QML. Concretely, we give
new diagrammatic proofs of the common 2- and 4-term parameter shift rules used
in the literature. Additionally, we derive a novel, generalised parameter shift
rule with 2n terms that is applicable to gates that can be represented with n
parametrised spiders in the ZXW-calculus. Furthermore, to the best of our
knowledge, we give the first proof of a conjecture by Anselmetti et al. by
proving a no-go theorem ruling out more efficient alternatives to the 4-term
shift rule.
  Secondly, we analyse the gradient landscape of quantum ans\"atze for barren
plateaus using both empirical and analytical techniques. Concretely, we develop
a tool that automatically calculates the variance of gradients and use it to
detect likely barren plateaus in commonly used quantum ans\"atze. Furthermore,
we formally prove the existence or absence of barren plateaus for a selection
of ans\"atze using diagrammatic techniques from the ZXW-calculus.
