Inserting Planar-Measured Qubits into MBQC Patterns while Preserving Flow

• URL: http://arxiv.org/abs/2508.14671v1
• Date: Wed, 20 Aug 2025 12:43:56 GMT
• Title: Inserting Planar-Measured Qubits into MBQC Patterns while Preserving Flow
• Authors: Miriam Backens, Thomas Perez,
• Abstract summary: Flow conditions are required for efficient translation into quantum circuits.<n>Most previous work has focused on rewrites that reduce the number of qubits in the computation.<n>We extend the definition of causal flow to permit YZ-measurements and derive the conditions under which a YZ-insertion preserves causal flow.
• Score: 1.4732811715354455
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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