  The one-way model of Measurement-Based Quantum Computing and the gate-based
circuit model give two different presentations of how quantum computation can
be performed. There are known methods for converting any gate-based quantum
circuit into a one-way computation, whereas the reverse is only efficient given
some constraints on the structure of the measurement pattern. Causal flow and
generalised flow have already been shown as sufficient, with efficient
algorithms for identifying these properties and performing the circuit
extraction. Pauli flow is a weaker set of conditions that extends generalised
flow to use the knowledge that some vertices are measured in a Pauli basis. In
this paper, we show that Pauli flow can similarly be identified efficiently and
that any measurement pattern whose underlying graph admits a Pauli flow can be
efficiently transformed into a gate-based circuit without using ancilla qubits.
We then use this relationship to derive simulation results for the effects of
graph-theoretic rewrites in the ZX-calculus using a more circuit-like data
structure we call the Pauli Dependency DAG.
