  Parameterized quantum circuits are attractive candidates for potential
quantum advantage in the near term and beyond. At the same time, as quantum
computing hardware not only continues to improve but also begins to incorporate
new features such as mid-circuit measurement and adaptive control,
opportunities arise for innovative algorithmic paradigms. In this work we focus
on measurement-based quantum computing protocols for approximate optimization,
in particular related to quantum alternating operator ans\"atze (QAOA), a
popular quantum circuit model approach to combinatorial optimization. For the
construction and analysis of our measurement-based protocols we demonstrate
that diagrammatic approaches, specifically ZX-calculus and its extensions, are
effective for adapting such algorithms to the measurement-based setting. In
particular we derive measurement patterns for applying QAOA to the broad and
important class of QUBO problems. We further outline how for constrained
optimization, hard problem constraints may be directly incorporated into our
protocol to guarantee the feasibility of the solution found and avoid the need
for dealing with penalties. Finally we discuss the resource requirements and
tradeoffs of our approach to that of more traditional quantum circuits.
