Inserting Planar-Measured Qubits into MBQC Patterns while Preserving Flow

2025-08-20

AI Summary

In measurement-based quantum computing (MBQC), a key challenge is inserting arbitrary planar measurements—i.e., measurements in the YZ or XZ planes, spanned by pairs of Pauli operators—into rewriting transformations (e.g., optimization or hardware mapping) while preserving flow conditions to ensure deterministic computation. This work generalizes causal flow, gflow, and Pauli flow to accommodate YZ- and XZ-plane measurements, establishing flow-preserving insertion criteria for such planar measurements and deriving corresponding vertex-splitting rules. By integrating these criteria with pivot transformations from the ZX-calculus, we achieve safe, deterministic insertion of arbitrary planar measurements, enabling universal quantum computation. Our framework extends the scope of flow conditions beyond the standard XY-plane, providing a theoretically rigorous and flexible foundation for MBQC pattern rewriting—particularly for circuit optimization, obfuscation, and hardware-aware compilation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Planning, Routing, and Scheduling: Deterministic Planning
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Extending causal flow to YZ-measurements in MBQC patterns
• Deriving conditions for planar-measured qubit insertion preserving flow
• Enabling flow-preserving rewriting for optimization and routing

Innovation

• Extended causal flow to YZ-measurements
• Derived conditions for flow-preserving qubit insertion
• Enabled universal MBQC with planar measurements