Complete Flow-Preserving Rewrite Rules for MBQC Patterns with Pauli Measurements

TL;DR

This paper introduces a complete set of graphical rewrite rules for measurement-based quantum computation patterns that preserve Pauli flow, enabling transformations that include adding qubits while maintaining determinism.

Contribution

It provides a novel, complete set of flow-preserving rewrite rules for MBQC patterns, including the addition of qubits, and establishes a canonical form for stabilizer ZX-diagrams.

Findings

Preserving Pauli flow during rewrites allows for more flexible pattern transformations.

Introduction of a canonical form for stabilizer ZX-diagrams with Pauli flow.

Complete reversible rewriting rules for MBQC patterns with Pauli flow.

Abstract

In the one-way model of measurement-based quantum computation (MBQC), computation proceeds via measurements on some standard resource state. So-called flow conditions ensure that the overall computation is deterministic in a suitable sense, with Pauli flow being the most general of these. Existing work on rewriting MBQC patterns while preserving the existence of flow has focused on rewrites that reduce the number of qubits. In this work, we show that introducing new Z-measured qubits, connected to any subset of the existing qubits, preserves the existence of Pauli flow. Furthermore, we give a unique canonical form for stabilizer ZX-diagrams inspired by recent work of Hu & Khesin. We prove that any MBQC-like stabilizer ZX-diagram with Pauli flow can be rewritten into this canonical form using only rules which preserve the existence of Pauli flow, and that each of these rules can be reversed while also preserving the existence of Pauli flow. Hence we have complete graphical rewriting for MBQC-like stabilizer ZX-diagrams with Pauli flow.