Inserting Planar-Measured Qubits into MBQC Patterns while Preserving Flow

In the one-way model of measurement-based quantum computation (MBQC), computation proceeds via single-qubit measurements on a resource state. Flow conditions ensure that the overall computation is deterministic in a suitable sense, and are required for efficient translation into quantum circuits. Procedures that rewrite MBQC patterns -- e.g. for optimisation, or adapting to hardware constraints -- thus need to preserve the existence of flow. Most previous work has focused on rewrites that reduce the number of qubits in the computation, or that introduce new Pauli-measured qubits. Here, we consider the insertion of planar-measured qubits into MBQC patterns, i.e. arbitrary measurements in a plane of the Bloch sphere spanned by a pair of Pauli operators; such measurements are necessary for universal MBQC. We extend the definition of causal flow, previously restricted to XY -measurements only, to also permit YZ-measurements and derive the conditions under which a YZ-insertion preserves causal flow. Then we derive conditions for YZ-insertion into patterns with gflow or Pauli flow, in which case the argument straightforwardly extends to XZ-insertions as well. We also show that the 'vertex splitting' or 'neighbour unfusion' rule previously used in the literature can be derived from YZ-insertion and pivoting. This work contributes to understanding the broad properties of flow-preserving rewriting in MBQC and in the ZX-calculus more broadly, and it will enable more efficient optimisation, obfuscation, or routing.