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In the one-way model of measurement-based quantum computation (MBQC),computation proceeds via measurements on a resource state. So-called flowconditions ensure that the overall computation is deterministic in a suitablesense, with Pauli flow being the most general of these. Computations,represented as measurement patterns, may be rewritten to optimise resource useand for other purposes. Such rewrites need to preserve the existence of flow toensure the new pattern can still be implemented deterministically. The majorityof existing work in this area has focused on rewrites that reduce the number ofqubits, yet it can be beneficial to increase the number of qubits for certainkinds of optimisation, as well as for obfuscation.  In this work, we introduce several ZX-calculus rewrite rules that increasethe number of qubits and preserve the existence of Pauli flow. These rules canbe used to transform any measurement pattern into a pattern containing only(general or Pauli) measurements within the XY-plane. We also give the firstflow-preserving rewrite rule that allows measurement angles to be changedarbitrarily, and use this to prove that the `neighbour unfusion' rule ofStaudacher et al. preserves the existence of Pauli flow. This implies it may bepossible to reduce the runtime of their two-qubit-gate optimisation procedureby removing the need to regularly run the costly gflow-finding algorithm.

Flow-preserving ZX-calculus Rewrite Rules for Optimisation and Obfuscation