There and back again: A circuit extraction tale

Giovanni De Felice; Hector Miller-Bakewell; John van de Wetering; Leo Lobski; Miriam Backens

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There and back again: A circuit extraction tale

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

Quantum Physics · Submitted 3 Mar 2020

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow.

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arXiv ID2003.01664 (version 3)

Pith IDOC2QWPBZ, Pith's permanent record of this paper and its review

CitationGiovanni De Felice, Hector Miller-Bakewell, John van de Wetering, Leo Lobski, Miriam Backens. "There and back again: A circuit extraction tale." arXiv:2003.01664 (2020). https://pith.science/paper/OC2QWPBZ

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What the abstract itself states, as probabilities: claims first 0.95 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.91 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Translations between the quantum circuit model and the measurement-based one-way model are useful for verification and optimisation of quantum computations. They make crucial use of a property known as gflow. While gflow is defined for one-way computations allowing measurements in three different planes of the Bloch sphere, most research so far has focused on computations containing only measurements in the XY-plane. Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow. The algorithm is efficient and the resulting circuits do not contain ancillae. One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams. In developing this algorithm, we generalise several concepts and results previously known for computations containing only XY-plane measurements. We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus. These rules are used to simplify measurement patterns by reducing the number of qubits while preserving both the semantics and the existence of gflow. The results can be applied to circuit optimisation by translating circuits to patterns and back again.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Translations between the quantum circuit model and the measurement-based one-way model are useful for verification and optimisation of quantum computations.	background 1.00	0.05	0.04	asserted 1.00
s1	They make crucial use of a property known as gflow.	background 1.00	0.05	0.03	asserted 1.00
s2	While gflow is defined for one-way computations allowing measurements in three different planes of the Bloch sphere, most research so far has focused on computations containing only measurements in the XY-plane.	gap 0.75	0.05	0.03	asserted 1.00
s3	Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow.	contribution 1.00	0.97	0.85	asserted 1.00
s4	The algorithm is efficient and the resulting circuits do not contain ancillae.	result 0.85	0.93	0.20	asserted 1.00
s5	One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams.	interpretation 0.84	0.69	0.18	asserted 1.00
s6	In developing this algorithm, we generalise several concepts and results previously known for computations containing only XY-plane measurements.	contribution 0.75	0.92	0.06	asserted 1.00
s7	We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus.	method 0.82	0.93	0.04	asserted 1.00
s8	These rules are used to simplify measurement patterns by reducing the number of qubits while preserving both the semantics and the existence of gflow.	method 0.98	0.70	0.07	asserted 1.00
s9	The results can be applied to circuit optimisation by translating circuits to patterns and back again.	interpretation 0.90	0.43	0.11	asserted 0.78

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

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arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Translations between the quantum circuit model and the measurement-based one-way model are useful for verification and optimisation of quantum computations. They make crucial use of a property known as gflow. While gflow is defined for one-way computations allowing measurements in three different planes of the Bloch sphere, most research so far has focused on computations containing only measurements in the XY-plane. Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow. The algorithm is efficient and the resulting circuits do not contain ancillae. One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams. In developing this algorithm, we generalise several concepts and results previously known for computations containing only XY-plane measurements. We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus. These rules are used to simplify measurement patterns by reducing the number of qubits while preserving both the semantics and the existence of gflow. The results can be applied to circuit optimisation by translating circuits to patterns and back again.

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References

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Finding gflow on unlabelled open graphs
quant-ph 2026-09 accept novelty 7.0 of 10

An O(n^3) algorithm finds a measurement labelling and compatible gflow on any unlabelled open graph, together with uniqueness and input-balancing results.
Entanglement entropy and magic of ZX-diagrams
quant-ph 2026-10 conditional novelty 6.0 of 10

Flow converts a ZX-diagram into a graph-state backbone dressed by non-Clifford gadgets, giving additive bounds on entanglement entropy (Eq. 2) and on logarithmic stabilizer extent (Eq. 4).

How later papers used it

48 later papers cite it

Evidence: substantial. 32 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 132 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 1 built on it and 2 used its method or tool.

Builds on it 1 of 2 papers with this label

Extended this paper's work on gflow, along with another work's, to handle a wider class of measurement patterns.

Relating Measurement Patterns to Circuits via Pauli Flow (2021) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 2 of 8 papers with this label

Implemented the algorithm described in Section 4.1 of this paper, adapting its operations on ZX diagrams to the measurement calculus.

Graphix: A software framework for Measurement-Based Quantum Computation (2026) · 0.97 · paraphrased: the licence does not allow quoting
The next step was extracting a new quantum circuit from the reduced ZX-graph [ref] .

Preparing Valence-Bond-Solid states on noisy intermediate-scale quantum computers (2022) · Basis Gate Decomposition of $e^-i π𝒮$ for Local Spin $S = 1, 3/2$ · 0.99

How the literature uses it

14 arXiv papers cite this in 32 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 22 sentences without a confident label

Builds on it 1 paper

In particular, we adapt the qubit gflow concept of `focusing' [ref] to and show that the existence of is equivalent to the existence of focused .

Working With Measurement-Based Computations on Qudits · Introduction · 0.96

Uses its method or tool 1 paper

This simplify-and-extract technique for ZX was introduced in [ref] , generalised to a broader family of diagrams in [ref] , and forms the basis of the quantum circuit optimisation tool PyZX [ref] (Fig. [ref]).

Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus) · Automated circuit optimisation · 0.89

Cites it as background 6 papers

Extracting quantum circuits back from graph-like ZX-diagrams where the circuit has only as many qubits as there are outputs/inputs, is so far only possible in polynomial time if the underlying graph has some kind of flow [ref] .

Multi-controlled Phase Gate Synthesis with ZX-calculus applied to Neutral Atom Hardware · Circuit extraction · 0.87
The process of converting a ZX-diagram into a quantum circuit is often referred to as circuit extraction [ref] .

Multi-agent blind quantum computation without universal cluster states · ZX-Calculus · 0.85
It has found use in quantum computation and quantum foundations, for instance in the fields of quantum circuit optimisation, verification and simulation, quantum error correction, and measurement-based quantum computation [ref] .

Spin-networks in the ZX-calculus · The ZXH-calculus · 0.88

What it cites · 12 sentences without a confident label

Builds on 2 papers

The extraction algorithm modifies that of Ref. [ref] so that it can handle measurements in multiples planes (not just the plane).

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus · Circuit extraction · 1.00
Our Lemmas [ref] and [ref] are generalisations of the work found in Ref. [ref] and Ref. [ref] .

Reducing T-count with the ZX-calculus · Removing Clifford vertices · 0.98

Uses the method or tool of 3 papers

Combined with the known procedure for transforming a quantum circuit into a measurement pattern using the [ref] , our pattern simplification and circuit extraction procedure can be used to reduce the T-gate count of quantum circuits.

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus · Our contributions · 0.90
A Jupyter notebook demonstrating the T-count optimisation is available at <https://github.com/Quantomatic/pyzx/blob/5d409a246857b7600cc9bb0fbc13043d54fb9449/demos/T-count%20Benchmark.ipynb>.] [ref] .

PyZX: Large Scale Automated Diagrammatic Reasoning · Conclusions and Future Work · 0.95
These measurement pattern rewrite rules, seen then as transformations of ZX-diagrams, were used in Ref. [ref] to reduce the T-count of circuits.

Reducing T-count with the ZX-calculus · Further pattern optimisation · 0.86

Compares with 1 paper

Such increases are avoided in the procedure of Ref. [ref] , where two-qubit gates are not resynthesized.

Reducing T-count with the ZX-calculus · Conclusions and Future Work · 0.88

Cites as background 4 papers

The `graph-like' studied in Ref. [ref] are MBQC+LC form diagrams where every vertex is measured in the plane.

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus · Removing Clifford vertices · 0.80
Phase gadgets have been used particularly in the study of circuit optimisation [ref] .

Reducing T-count with the ZX-calculus · Phase-gadget form · 0.99
Phase gadgets have been used particularly in the study of circuit optimisation [ref] .

Phase Gadget Synthesis for Shallow Circuits · Phase-gadget form · 0.99

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