Generating one-way computations with flow: flow-preserving rewriting that ignores the interpretation

Abstract: The one-way model is a universal model of quantum computation, driven by successive adaptive single-qubit measurements on an entangled resource state. Measurements are non-deterministic, yet if the computation satisfies one of several related families of conditions known as 'flows', the computation can be made deterministic overall by modifying later measurements depending on the outcomes of earlier ones. Flow properties also enable efficient translation from one-way computations to circuits, motivating research into rewriting one-way computations while preserving the existence of flow. Existing approaches to flow-preserving rewriting are used for compilation or optimisation and preserve both the interpretation and the existence of flow. Here, we broaden our perspective to consider flow-preserving rewriting that does not necessarily preserve the interpretation, with applications to creating test instances for software that works with flow, as well as to generating ansätze for quantum machine learning. We show that a family of just three flow-preserving rewrite rules suffices to generate any diagram with flow from a trivial diagram with the desired number of inputs and outputs. This rule set is nearly the same as the complete set of flow- and interpretation-preserving rewrite rules for one-way computations in which all measurements are Pauli; and just a small subset of the flow- and interpretation-preserving rewrite rules for arbitrary measurements.

• The paper establishes a minimal, complete set of three flow-preserving rewrite rules that generate any MBQC instance with flow from a trivial diagram.
• It leverages the ZX-calculus and graph theory to manage input/output wiring and local complementations crucial for preserving the flow structure.
• The work demonstrates algorithmic techniques for generating MBQC patterns and offers insights for software testing and quantum variational ansatz design.

Flow-Preserving Rewrite Rules for One-Way Quantum Computations

Introduction

The paper "Generating one-way computations with flow: flow-preserving rewriting that ignores the interpretation" (2607.03250) focuses on the theoretical foundations and construction methods for generating one-way measurement-based quantum computation (MBQC) patterns that preserve flow but not necessarily the semantic interpretation. The analysis is situated within the ZX-calculus and graph-theoretic frameworks, and develops a minimal, complete set of flow-preserving rewrite rules—demonstrating that this set suffices to generate any MBQC instance with flow from a trivial diagram. The work clarifies the structural mechanisms behind deterministic MBQC, algorithmic MBQC instance generation, and the separation of flow properties from computational equivalence.

Background: Flow in One-Way MBQC

In one-way quantum computation, adaptive single-qubit measurements are performed on an entangled resource state—typically a graph state. Measurement-induced nondeterminism is counteracted by the presence of a "flow" property, a partial order with associated correction sets, that prescribes how corrective operations can be scheduled to ensure deterministic output. Different flow variants (causal flow, gflow, Pauli flow) address various resource states, measurement settings, and optimization constraints.

The ZX-calculus provides a graphical formalism bridging MBQC and circuit models, and supports graphical manipulation via equational reasoning. However, the full ZX-calculus' equational theory, while complete for circuit equivalence, often does not preserve MBQC-specific structures like flow, motivating the search for restricted rule sets that maintain flow during rewriting operations.

Main Results: Minimal Flow-Preserving Rewrite Set

The central contribution is the identification of a minimal, complete set of three flow-preserving rewrite rules in the ZX-calculus/graphical MBQC framework. Unlike previous rule sets aimed at compilation/optimization, these rules do not necessarily preserve the computational meaning ("interpretation") of the MBQC pattern, but only the structural property of flow. This focus is justified by the need to generate MBQC patterns for testing, benchmarking, and variational quantum circuit synthesis.

The three key rules are:

1. Input/Output Wire Extension/Contraction ("IO"): Allows inserting/removing degree-two vertices along paths to or from boundary vertices. This operation preserves flow and can manipulate the input/output wiring scheme.
2. Local Complementation (LC): Complements the neighborhood of a given vertex (not an input), corresponding to the local Clifford group action. LC manipulates entanglement without altering flow existence and, in conjunction with other operations, can realize pivots and more complex graph reshaping.
3. Z-like Measurement Insertion/Deletion (ZL): Insertion or removal of single qubit measurements in the Z, XZ, or YZ planes ("Z-like" vertices). Deletion is always flow-preserving, whereas insertion is flow-preserving only under nontrivial, structurally characterized conditions. Notably, the insertion rule's side conditions involve analysis of the graph's correction sets and partial order structure, as prescribed by the algebraic flow characterization.

The set is minimal: omitting any rule sacrifices universality for MBQC with flow—IO is necessary for input/output boundary management, ZL is required for connected component manipulation, and LC is essential for generating nontrivial measurement bases and altering the ambient planar graph structure.

Structural Theorems and Generation Algorithms

The analysis provides constructive completeness results: any MBQC pattern with Pauli flow can be reduced, via these rules, to a trivial normal form (an identity map with matching numbers of inputs/outputs, no edges, and only output vertices). The proof strategy involves sequential elimination of Pauli measurements, followed by elimination of Z-like planar measurements, and then iterative merging of non-output vertices into outputs. The output permutation can be achieved by a bounded sequence of rewrites, leveraging the structural properties of the IO and LC rules.

Algorithmic inversion of the trivialization process enables the generation of arbitrary MBQC instances with flow, by iteratively applying flow-preserving insertions in reverse. The rule set is also sufficient for generating all gflow-preserving instances (planar measurements), and, with small extension, all-XY measurement diagrams relevant for universal MBQC.

The construction is compatible with the algebraic formulation of flow (based on adjacency, flow-demand, and order-demand matrices), enabling explicit tracking and verification of flow preservation during rewriting and algorithmic instance generation.

Derived Rules and Algorithmic Implications

The paper identifies several derived transformations (edge toggling, neighbor merging, vertex splitting, and pivot-and-delete) that, while not atomic in the rule set, are synthesized from sequences of the three basic rules. These derived forms clarify critical diagrammatic manipulations (such as cluster state modifications and measurement plane scheduling) and are leveraged for efficient diagram generation and analysis.

A key observation is that, in the output-localized generation algorithm, the number of choices at each step depends only on the current number of outputs, not the total qubit count, providing a scalable approach for diagram generation. The combinatorics of partial orderings underpin the sampling distribution over generated MBQC instances.

Furthermore, the work highlights that flow preservation is structurally orthogonal to computational equivalence (interpretation), as evidenced by the similarity of the presented minimal rule set to rule sets for the Clifford fragment and the central role of phase management in interpretation-preserving rewrites.

Implications and Future Directions

The clarified structural mechanics of flow-preserving rewriting have practical relevance for software testing in quantum programming frameworks (e.g., Graphix), automatic generation of MBQC patterns with guaranteed determinism, and the design of quantum variational ansätze that exploit measurement adaptivity.

Theoretically, the work suggests a clear separation between topology-changing, flow-critical rules and phase-management rules in the ZX/MBQC landscape. This demarcation enables targeted analysis of MBQC universality, resource state architecture, and compilation pathways, particularly for quantum machine learning architectures and automated search spaces.

The established minimality provides a foundation for the complexity-theoretic analysis of MBQC instance generation and suggests directions for further research into distributional properties and constraint satisfaction within MBQC flow spaces.

Finally, the algorithmic techniques introduced—both for simplifying arbitrary flow-preserving diagrams and for systematically constructing flow-rich instances—provide a toolkit for both theoretical exploration and experimental workflow design.

Conclusion

This work formalizes the minimal and sufficient conditions for generating arbitrary one-way quantum computations with flow, independent of computational equivalence. The identification and analysis of a three-rule, flow-preserving rewrite set in the ZX-calculus offers both theoretical insight and practical utility—enabling algorithmic generation, software testing, and architectural search for deterministic MBQC. The separation of flow structure from interpretation management refines the conceptual architecture of MBQC rewriting, and highlights avenues for future algorithmic and foundational investigation.