Completeness for flow-preserving rewrite rules

Miriam Backens; Simon Perdrix

REVIEW 2 major objections 4 minor 12 references

Completeness for flow-preserving rewrite rules

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The flow-preserving rewrite rules of Figure 2 are complete: any two equivalent MBQC-form ZX-diagrams with Pauli flow can be transformed into each other while keeping a flow.

desk verdict A strong, careful completeness proof for flow-preserving ZX rewriting that reduces to circuit completeness; the reduction has a genuine gap around the reverse of the spider-nest rule, but the main theorem is plausible and worth refereeing. read the letter →

arxiv 2608.13035 v1 pith:YOF7VDSR submitted 2026-08-13 quant-ph

Miriam Backens , Simon Perdrix This is my paper · ORCID

classification quant-ph

keywords ZX-calculusmeasurement-basedquantumcomputingPauliflowgflowcausalcircuitextractionflow-preservingrewritingcircuitswithinitialisation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the rewrite rules of Figure 2 are complete for flow-preserving transformations of measurement-based quantum computations: any two MBQC-form ZX-diagrams that both have Pauli flow and represent the same linear map can be rewritten into each other using only rules that preserve the existence of a flow. This extends the previously known complete flow-preserving calculus for the Clifford fragment to the full Pauli-flow fragment, and it is the first time quantum-circuit completeness results are used to prove a ZX-calculus completeness theorem. The proof works by making circuit extraction itself flow-preserving and then translating a complete equational theory for quantum circuits with initialisation into ZX rewrites. The result matters because Pauli flow is the standard certificate that a measurement-based computation is deterministic and can be efficiently turned into a quantum circuit, so a complete flow-preserving calculus lets one reason graphically without ever leaving the class of efficiently extractable diagrams.

What carries the argument

The central machinery is the triple of flow notions — Pauli flow, gflow, and extended causal flow — organised by correction sets that are subsets of the inputs. The load-bearing construction is 'extracted vertices' and their frontier (Definition 43): a growing set at the top of the dependency order whose correction sets have size one, with a frontier that is always as large as the number of outputs. Each extraction step applies YZ-insertion followed by a pivot, mimicking a CNOT or Hadamard push into the circuit while preserving the existence of gflow; once all vertices are extracted, correction sets have size one and the diagram has extended causal flow, meaning every non-output vertex is corrected by a single neighbouring vertex. Around this, the causal-subdiagram lemma lets a rewrite inside a small subdiagram be replaced in any context without destroying extended causal flow, and the spider-nest rule removes full phase-gadget nests that implement a trivial 2π phase.

What would settle it

Run an exhaustive search over all MBQC-form ZX-diagrams up to a small size that admit Pauli flow, grouping them by their linear map; if any two in the same group are not connected by the Figure 2 rules applied under their stated flow conditions, Theorem 74 is false. A cheaper check targets the imported circuit completeness: verify the Figure 5 rules by direct matrix computation on all circuits with, say, up to three qubits and a bounded number of gates.

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Extended reading notes

Core claim

The central claim is Theorem 74: the flow-preserving rule set of Figure 2 — input/output unfusion, phase fusion, Z-insertion/deletion, local complementation, Euler decomposition, and the spider-nest rule — is complete for MBQC-form ZX-diagrams that admit Pauli flow. In concrete terms, whenever two such diagrams denote the same linear embedding, there is a sequence of these rewrites connecting them, and every intermediate diagram also has a flow. The proof has two stages. First, a manifestly flow-preserving circuit-extraction algorithm turns any diagram with Pauli flow into a diagram with extended causal flow, using YZ-insertions and pivots that never increase the number of unextracted vertices and maintain a pseudo-focused gflow. Second, the paper shows that the ZX-translations of a complete set of rewrite rules for quantum circuits with initialisation are derivable from Figure 2; a causal-subdiagram lemma (Definition 64, Proposition 71) allows those derivations to be lifted from local contexts to arbitrary diagrams. The completeness of the flow-preserving calculus therefore reduces to the completeness of circuit rewriting.

Load-bearing premise

The main theorem inherits the completeness of the imported equational theory for quantum circuits with qubit initialisation; if that circuit-completeness theorem has any flaw, the ZX completeness result collapses with it.

Editorial extensions

If this is right

Any two equivalent MBQC-form ZX-diagrams with Pauli flow are connected by flow-preserving rewrites; no deterministic measurement-based computation needs to leave the flow-preserving fragment to be compared with another.
Circuit extraction becomes a diagrammatic derivation: every diagram with Pauli flow can be rewritten, using only the Figure 2 rules, into an extended-causal-flow diagram that is directly readable as a quantum circuit with initialisation, with at most O(n^2) new vertices.
The non-Clifford part of the calculus is handled by the Euler decomposition rule and the spider-nest rule, which together absorb all real phase angles; this is what extends the earlier Clifford-completeness result.
Any equality between two circuits with initialisation can be proved in the ZX-calculus without ever destroying the flow, because every circuit rewrite of Figure 5 has a flow-preserving ZX counterpart.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The reduction makes the main theorem conditional on the cited circuit-completeness results; independently re-checking those proofs, or formalising them, would directly test the foundation of Theorem 74.
A similar recipe could yield flow-preserving completeness elsewhere: choose a complete circuit calculus and find flow-preserving ZX translations for its rules, as done here for initialisation; qudit MBQC or diagrams with ZX-flow are natural next targets.
The spider-nest identity suggests a concrete optimisation heuristic: search for full phase-gadget nests whose phases sum to 2π and delete them; because the rule preserves flow, such optimisation could be safely interleaved with circuit extraction.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a serious paper and likely the right result: complete flow-preserving rewrites for MBQC-form ZX-diagrams with Pauli flow. The new rules (PF, EZX, SN) and the extraction argument are real contributions. The proof that Pauli flow can be brought to extended causal flow while preserving flow is detailed and convincing; I read the diagram lemmas carefully and the flow bookkeeping checks out.

What is actually new: previous completeness stopped at the Clifford fragment, and this extends it to Pauli flow. The strategy of using quantum circuit completeness as the engine, instead of proving ZX completeness directly, is novel. Sections 3 and 4 are the strongest part; the extraction algorithm is carefully argued and genuinely manifestly flow-preserving in a way I have not seen before.

The soft spot is the handoff in Section 5. Theorem 74 leans on Theorem 59, which imports completeness from [7,8,9] with only a sketch. That is not fatal by itself, but the translation of circuit derivations into ZX rewrites is not fully demonstrated. In particular, rule (I) is translated only in the deleting direction via SN; the reverse of (I) would require adding a full 2π spider nest, and the reverse of SN is never proved. The paper says only the deletion direction is needed, but the circuit completeness proof explicitly reverses every rule, including (I). The derivation of reverse (I) in the circuit calculus introduces a 4π gate and then removes one 2π copy; translating that seems to need either reverse SN or a derivation the paper does not supply. For n≥4 the phases are not Clifford, so the Clifford-complete rules will not cover it. This is a genuine gap in the written proof, not a fatal flaw — I suspect reverse SN is true, or the derivation can be rearranged, but it needs to be shown.

Observation 65, the context-lifting claim, is also asserted rather than proved. The argument is plausible and probably fixable, but it is load-bearing for the claim that local circuit rewrites lift to arbitrary contexts.

Bottom line: the paper is for people working on ZX-calculus completeness and MBQC circuit optimization. It deserves a serious referee, but the referee should ask for a complete proof of the circuit-to-ZX translation, especially the (I) direction, before acceptance.

Referee Report

2 major / 4 minor

Summary. The paper proves a completeness theorem for a set of flow-preserving ZX-calculus rewrite rules (Figure 2). The main result, Theorem 74, states that any two MBQC-form ZX-diagrams that both have Pauli flow and represent the same linear embedding can be transformed into each other using only rules that preserve the existence of flow. The proof has two parts. Section 4 gives a flow-preserving circuit extraction algorithm, showing that any MBQC-form diagram with Pauli flow can be rewritten, using only the Figure 2 rules, into a diagram with extended causal flow; this is done via a phase-gadget form and a notion of pseudo-focused gflow, with explicit correction-function updates. Section 5 reduces completeness for extended-causal-flow diagrams to a completeness theorem for quantum circuits with initialisation (Theorem 59). The ZX-translations of the circuit rules are shown to be derivable from Figure 2 (Proposition 73), and a causal-subdiagram formalism (Proposition 71) is introduced to lift local rewrites to arbitrary contexts. New ingredients include the phase-fusion rule (PF), the Euler rule (EZX), and the spider-nest rule (SN).

Significance. If the main theorem is correct, this is a significant advance: it extends complete flow-preserving rewriting from the Clifford fragment to all MBQC-form diagrams with Pauli flow, and it provides a manifestly flow-preserving circuit extraction procedure that may be useful for circuit optimization. The paper is careful and detailed: correction-function updates are given explicitly in Lemmas 52 and 55, the pseudo-focused flow machinery is well developed, and the paper honestly states which rule directions are conditional. The central claim, however, currently rests on two under-supported steps: the reverse direction of the spider-nest rule (SN) is not proved, and Observation 65, which lifts circuit rewrites to causal subdiagrams, is asserted rather than proved. These are fixable, but they are load-bearing for Theorem 74, so the result is not yet established as written.

major comments (2)

[§3.2 (Lemma 28) and §5.3 (Lemma 72)] Theorem 74 simulates circuit derivations that may use rule (I) of Figure 5 in reverse, because Theorem 59 proves that every rule of Figure 5 can be reversed. However, Lemma 28 proves only the deletion direction of the spider-nest rule (SN), and Lemma 72 only derives the forward direction of (I) by applying (SN). The reverse of (I) would require adding a full 2π spider nest, i.e. the reverse of (SN), which the paper never shows to be flow-preserving. Lemma 36 handles n≤3 using Clifford rules, but for n≥4 the nest contains phases that are not multiples of π/2, so that argument does not apply. Consequently the reduction from circuit completeness to ZX completeness is incomplete at exactly the step where a circuit derivation uses (I) in reverse; this is load-bearing for Theorem 74.
[§5.2, Observation 65] Observation 65 asserts, without proof, that any subcircuit to which a circuit rewrite rule can be applied translates to a causal subdiagram in the sense of Definition 64. This is load-bearing: Theorem 74 uses it to conclude that any sequence of circuit rewrites lifts to a sequence of valid flow-preserving ZX rewrites via Proposition 71. The surrounding text gives intuition and one example, but no proof is given that Conditions 1–4 of Definition 64 always hold, including for rules such as the right-to-left direction of (C) which act on possibly disconnected parts, and for circuits with initialisation where the 'past' may be empty. A formal construction of Vpast, VD, and Vfuture for each circuit rewrite rule, or a proof by induction on circuit structure, should be supplied.

minor comments (4)

[§5.1, Theorem 59] The completeness proof imports the core unitary-circuit completeness from [7] and the initialisation extension from [8]. Please state the specific theorem numbers and explicitly justify that the theory from [8], which allows both initialisation and termination, specialises to the initialisation-only setting used here; since Theorem 74 inherits this result, a precise statement would help the reader.
[Definition 43] The line 'We sometimes refer to S:=V\S as the set of unextracted vertices' defines a set as its own complement because S is already the set of extracted vertices; use an overline or a different letter for the complement.
[Algorithm 1] Line 5 introduces the variable T for the set of non-extracted vertices, while the surrounding text and Lemmas 52–55 use S for the same concept; aligning these notations would avoid confusion.
[Lemma 32] The right-to-left condition reads 'extended causal flow is preserved if The conditions...', which appears to be a typographical error; the sentence should be rephrased.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main theorem reduces flow-preserving ZX completeness to independent circuit-completeness results; the reverse-(SN) issue is a proof gap, not a circular step.

full rationale

The central derivation chain of Theorem 74 is a reduction rather than a re-labelling of its inputs. The proof first turns any MBQC-form ZX-diagram with Pauli flow into an extended-causal-flow diagram using the flow-preserving rules of Figure 2 (Lemmas 42, 52, 55 and Algorithm 1), and then imports Theorem 59, a completeness result for quantum circuits with initialisation, to connect the resulting circuit-like diagrams. Theorem 59 is cited to Refs [7,8,9]; although Perdrix is a co-author of those papers, they are separate published results with their own proofs and do not assume flow-preserving ZX completeness. No parameter is fitted and no quantity is defined in terms of the target completeness statement, so the reduction is not circular by construction. The one flagged weakness is an omitted proof rather than circularity: Lemma 28 proves only the deletion direction of the spider-nest rule (SN), and Lemma 72 uses (SN) only left-to-right to derive the ZX-translation of circuit rule (I). Since the circuit completeness proof of Theorem 59 explicitly reverses every rule, the reverse of (I) would appear to require the reverse of (SN) (or an alternative derivation) when translated to ZX, and that reverse direction is not supplied. This is a potential correctness gap in the reduction, but it is not a self-referential or definitional equivalence, so it does not increase the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure mathematics proof with no fitted parameters and no new physical entities. It imports several prior results as axioms, most notably the completeness of quantum circuit equational theories [7, 8, 9], the standard flow-finding algorithms, and the phase-gadget Fourier correspondence. No circular reliance on the main theorem being proved.

assumptions (4)

domain assumption Standard ZX-calculus semantics: the interpretation of ZX-diagrams as linear maps and the soundness of the diagrammatic equations of Figure 1.
The paper assumes the standard ZX-calculus interpretation and uses standard ZX rewrites to prove interpretation preservation of the new rules (e.g., Lemma 24). Section 2.1.
domain assumption Labelled open graphs and flow conditions (Pauli flow, gflow, causal flow) are defined as in [6, 11, 29].
Definitions 2 to 10; the paper builds on these established notions without reproving their basic properties.
domain assumption Completeness of the equational theory for quantum circuits with initialisation, Theorem 59, imported from Refs [7, 8, 9].
The proof of Theorem 59 sketches the reduction but relies on completeness proofs in [7, 8, 9]; the central Theorem 74 depends on this.
domain assumption The Fourier transform correspondence between phase gadgets and multi-controlled phase gates, cf. [18, Proposition 3.6].
Used in Lemma 28 (spider nest rule) to show a full spider nest of phase 2pi implements the identity.

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Cite this review

Pith. "Pith review of Completeness for flow-preserving rewrite rules." pith.science (2026). https://pith.science/paper/YOF7VDSR

@misc{pith2026260813035,
  author       = {Pith},
  title        = {Pith review of: Completeness for flow-preserving rewrite rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOF7VDSR}},
  note         = {Machine review of arXiv:2608.13035}
}

read the original abstract

Complete sets of graphical rewrite rules enable fully graphical reasoning about quantum computations and have been an area of active research for more than a decade. Many recent applications of the ZX-calculus have made use of the close correspondence between ZX-diagrams and computations in the one-way model of measurement-based quantum computation. In this model, various kinds of flow properties ensure deterministic implementability; for ZX-diagrams, these same properties allow efficient translation into quantum circuits (a problem that is known to be #P-hard in general). Therefore, flow-preserving ZX-calculus rewrite rules are of strong interest. Here, we extend the set of flow-preserving rules appearing in the literature with a few new rules and extensions of existing rules. We then show that the resulting rule set is complete for all flow-preserving translations between ZX-diagrams of appropriate form. The proof employs a manifestly flow-preserving equivalent of circuit extraction, where a diagram with gflow is transformed, using only flow-preserving rewrite rules, into a diagram with causal flow.

Figures

Figures reproduced from arXiv: 2608.13035 by the authors.

**Figure 2.** Rules (PF) and (LC) are stated as directed rules because they provide their own inverses: in the former case this is by choosing α ′ suitably, in the latter case it is because repeating the rule four times implements an overall identity transformation. Note that (EZX) has been proved specifically in the context of (extended) causal flow, which indeed is the only situation where it will be needed. Conversely, rule (L… view at source ↗

**Figure 2.** ▶ Lemma 40 (Phase gadget addition). Let Γ = (G, I, O, λ) be a labelled open graph with extended causal flow g. Suppose there exist two Y Z-measured vertices a, b ∈ O such that NG(a) = NG(b). Then these two phase gadgets can be replaced by a single phase gadget whose phase is the sum of the original two phases, or conversely: α . . . . . . . . . . . . β ⇄ α + β . . . . . . . . . . . . Proof. Recall that in an extende… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Graphical Fourier Theory and the Cost of Quantum Addition

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [5]

doi:10.4230/LIPIcs.CSL.2024.20

Schloss Dagstuhl – Leibniz-Zentrum für Informatik. doi:10.4230/LIPIcs.CSL.2024.20. 9 Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron. A Complete Equational Theory for Quantum Circuits. In38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1–13, Boston, MA, USA, June

doi:10.4230/lipics.csl.2024.20 2024

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28 Robert Raussendorf and Hans J. Briegel. A One-Way Quantum Computer.Physical Review Letters, 86(22):5188–5191, May 2001.doi:10.1103/PhysRevLett.86.5188. 29 Will Simmons. Relating Measurement Patterns to Circuits via Pauli Flow.Electronic Proceedings in Theoretical Computer Science, 343:50–101, September 2021.doi:10.4204/ EPTCS.343.4. 30 Korbinian Stauda...

arXiv 2001

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Working with measurement-based computations on qudits

doi:10.1007/ 978-3-540-70575-8_70. 24 Piotr Mitosek and Miriam Backens. An algebraic formulation of Pauli flow, leading to faster flow-finding algorithms.Journal of Physics A: Mathematical and Theoretical, 59(3):035301, January 2026.doi:10.1088/1751-8121/ae2999. 25 Piotr Mitosek and Miriam Backens. Working with measurement-based computations on qudits, Ju...

work page Pith review arXiv

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11 Vincent Danos and Elham Kashefi

doi:10.1088/1367-2630/ 13/4/043016. 11 Vincent Danos and Elham Kashefi. Determinism in the one-way model.Physical Review A, 74(5):052310, November 2006.doi:10.1103/PhysRevA.74.052310. 12 N. de Beaudrap. Finding flows in the one-way measurement model.Physical Review A, 77(2):022328, February 2008.doi:10.1103/PhysRevA.77.022328. 50 REFERENCES 13 N. de Beaud...

doi:10.1088/1367-2630/ 2006

[5] [2014]

2 Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, and John van de Wetering

doi:10.1088/1367-2630/16/9/ 093021. 2 Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, and John van de Wetering. There and back again: A circuit extraction tale.Quantum, 5:421, March

work page doi:10.1088/1367-2630/16/9/

[6] [2017]

doi:10.1007/978-3-662-55751-8_31

Springer. doi:10.1007/978-3-662-55751-8_31. REFERENCES 51 27 Thomas Perez. Measurement-Based Quantum Computing : Inserting YZ-measured qubits in MBQC patterns while preserving flow conditions. M1 Internship Report, University of Birmingham,

work page doi:10.1007/978-3-662-55751-8_31

[7] [2020]

Graphical Fourier Theory and the Cost of Quantum Addition

Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.doi:10.22331/q-2020-06-04-279. 15 Ross Duncan and Simon Perdrix. Rewriting Measurement-Based Quantum Computa- tions with Generalised Flow. In Samson Abramsky, Cyril Gavoille, Claude Kirchner, Friedhelm Meyer auf der Heide, and Paul G. Spirakis, editors,Automata, Lang...

work page Pith review arXiv 2020

[8] [2021]

3 Miriam Backens and Thomas Perez

Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.doi:10.22331/q-2021-03-25-421. 3 Miriam Backens and Thomas Perez. Inserting Planar-Measured Qubits into MBQC Patterns while Preserving Flow.Electronic Proceedings in Theoretical Computer Science, 426:100–126, August 2025.doi:10.4204/EPTCS.426.4. 4 Robert Ivan Booth.M...

doi:10.22331/q-2021-03-25-421 2021