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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Although quantum circuits have been ubiquitous for decades in quantum computing, the first complete equational theory for quantum circuits has only recently been introduced. Completeness guarantees that any true equation on quantum circuits can be derived from the equational theory.
We improve this completeness result in two ways: (i) We simplify the equational theory by proving that several rules can be derived from the remaining ones. In particular, two out of the three most intricate rules are removed, the third one being slightly simplified. (ii) The complete equational theory can be extended to quantum circuits with ancillae or qubit discarding, to represent respectively quantum computations using an additional workspace, and hybrid quantum computations. We show that the remaining intricate rule can be greatly simplified in these more expressive settings, leading to equational theories where all equations act on a bounded number of qubits.
The development of simple and complete equational theories for expressive quantum circuit models opens new avenues for reasoning about quantum circuits. It provides strong formal foundations for various compiling tasks such as circuit optimisation, hardware constraint satisfaction and verification.

Alexandre Clément, Noé Delorme, Simon Perdrix, and Renaud Vilmart. Quantum Circuit Completeness: Extensions and Simplifications. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{clement_et_al:LIPIcs.CSL.2024.20, author = {Cl\'{e}ment, Alexandre and Delorme, No\'{e} and Perdrix, Simon and Vilmart, Renaud}, title = {{Quantum Circuit Completeness: Extensions and Simplifications}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {20:1--20:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.20}, URN = {urn:nbn:de:0030-drops-196639}, doi = {10.4230/LIPIcs.CSL.2024.20}, annote = {Keywords: Quantum Circuits, Completeness, Graphical Language} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We exhibit a strong connection between the matchgate formalism introduced by Valiant and the ZW-calculus of Coecke and Kissinger. This connection provides a natural compositional framework for matchgate theory as well as a direct combinatorial interpretation of the diagrams of ZW-calculus through the perfect matchings of their underlying graphs.
We identify a precise fragment of ZW-calculus, the planar W-calculus, that we prove to be complete and universal for matchgates, that are linear maps satisfying the matchgate identities. Computing scalars of the planar W-calculus corresponds to counting perfect matchings of planar graphs, and so can be carried in polynomial time using the FKT algorithm, making the planar W-calculus an efficiently simulable fragment of the ZW-calculus, in a similar way that the Clifford fragment is for ZX-calculus. This work opens new directions for the investigation of the combinatorial properties of ZW-calculus as well as the study of perfect matching counting through compositional diagrammatical technics.

Titouan Carette, Etienne Moutot, Thomas Perez, and Renaud Vilmart. Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 120:1-120:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{carette_et_al:LIPIcs.ICALP.2023.120, author = {Carette, Titouan and Moutot, Etienne and Perez, Thomas and Vilmart, Renaud}, title = {{Compositionality of Planar Perfect Matchings: A Universal and Complete Fragment of ZW-Calculus}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {120:1--120:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.120}, URN = {urn:nbn:de:0030-drops-181726}, doi = {10.4230/LIPIcs.ICALP.2023.120}, annote = {Keywords: Perfect Matchings Counting, Quantum Computing, Matchgates, ZW-Calculus, String Diagrams, Completeness} }

Document

**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

The "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems, and is a tool that is used in formal verification of such systems.
We give here a new set of rewrite rules for the formalism, and show that it is complete for "Toffoli-Hadamard", the simplest approximately universal fragment of quantum mechanics. We show that the rewriting is terminating, but not confluent (which is expected from the universality of the fragment). We do so using the connection between Sum-over-Paths and graphical language ZH-Calculus, and also show how the axiomatisation translates into the latter.
Finally, we show how to enrich the rewrite system to reach completeness for the dyadic fragments of quantum computation - obtained by adding phase gates with dyadic multiples of π to the Toffoli-Hadamard gate-set - used in particular in the Quantum Fourier Transform.

Renaud Vilmart. Completeness of Sum-Over-Paths for Toffoli-Hadamard and the Dyadic Fragments of Quantum Computation. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 36:1-36:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{vilmart:LIPIcs.CSL.2023.36, author = {Vilmart, Renaud}, title = {{Completeness of Sum-Over-Paths for Toffoli-Hadamard and the Dyadic Fragments of Quantum Computation}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {36:1--36:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.36}, URN = {urn:nbn:de:0030-drops-174974}, doi = {10.4230/LIPIcs.CSL.2023.36}, annote = {Keywords: Quantum Computation, Verification, Sum-Over-Paths, Rewrite Strategy, Toffoli-Hadamard, Completeness} }

Document

**Published in:** LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Recent developments in classical simulation of quantum circuits make use of clever decompositions of chunks of magic states into sums of efficiently simulable stabiliser states. We show here how, by considering certain non-stabiliser entangled states which have more favourable decompositions, we can speed up these simulations. This is made possible by using the ZX-calculus, which allows us to easily find instances of these entangled states in the simplified diagram representing the quantum circuit to be simulated. We additionally find a new technique of partial stabiliser decompositions that allow us to trade magic states for stabiliser terms. With this technique we require only 2^{α t} stabiliser terms, where α≈ 0.396, to simulate a circuit with T-count t. This matches the α found by Qassim et al. [Qassim et al., 2021], but whereas they only get this scaling in the asymptotic limit, ours applies for a circuit of any size. Our method builds upon a recently proposed scheme for simulation combining stabiliser decompositions and optimisation strategies implemented in the software QuiZX [Kissinger and van de Wetering, 2022]. With our techniques we manage to reliably simulate 50-qubit 1400 T-count hidden shift circuits in a couple of minutes on a consumer laptop.

Aleks Kissinger, John van de Wetering, and Renaud Vilmart. Classical Simulation of Quantum Circuits with Partial and Graphical Stabiliser Decompositions. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 5:1-5:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kissinger_et_al:LIPIcs.TQC.2022.5, author = {Kissinger, Aleks and van de Wetering, John and Vilmart, Renaud}, title = {{Classical Simulation of Quantum Circuits with Partial and Graphical Stabiliser Decompositions}}, booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)}, pages = {5:1--5:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-237-2}, ISSN = {1868-8969}, year = {2022}, volume = {232}, editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.5}, URN = {urn:nbn:de:0030-drops-165128}, doi = {10.4230/LIPIcs.TQC.2022.5}, annote = {Keywords: ZX-calculus, Stabiliser Rank, Quantum Simulation} }

Document

**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

ZX-Calculus is a versatile graphical language for quantum computation equipped with an equational theory. Getting inspiration from Geometry of Interaction, in this paper we propose a token-machine-based asynchronous model of both pure ZX-Calculus and its extension to mixed processes. We also show how to connect this new semantics to the usual standard interpretation of ZX-diagrams. This model allows us to have a new look at what ZX-diagrams compute, and give a more local, operational view of the semantics of ZX-diagrams.

Kostia Chardonnet, Benoît Valiron, and Renaud Vilmart. Geometry of Interaction for ZX-Diagrams. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 30:1-30:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chardonnet_et_al:LIPIcs.MFCS.2021.30, author = {Chardonnet, Kostia and Valiron, Beno\^{i}t and Vilmart, Renaud}, title = {{Geometry of Interaction for ZX-Diagrams}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {30:1--30:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.30}, URN = {urn:nbn:de:0030-drops-144701}, doi = {10.4230/LIPIcs.MFCS.2021.30}, annote = {Keywords: Quantum Computation, Linear Logic, ZX-Calculus, Geometry of Interaction} }

Document

**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Graphical calculi such as the ZH-calculus are powerful tools in the study and analysis of quantum processes, with links to other models of quantum computation such as quantum circuits, measurement-based computing, etc.
A somewhat compact but systematic way to describe a quantum process is through the use of quantum multiple-valued decision diagrams (QMDDs), which have already been used for the synthesis of quantum circuits as well as for verification.
We show in this paper how to turn a QMDD into an equivalent ZH-diagram, and vice-versa, and show how reducing a QMDD translates in the ZH-Calculus, hence allowing tools from one formalism to be used into the other.

Renaud Vilmart. Quantum Multiple-Valued Decision Diagrams in Graphical Calculi. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 89:1-89:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{vilmart:LIPIcs.MFCS.2021.89, author = {Vilmart, Renaud}, title = {{Quantum Multiple-Valued Decision Diagrams in Graphical Calculi}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {89:1--89:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.89}, URN = {urn:nbn:de:0030-drops-145295}, doi = {10.4230/LIPIcs.MFCS.2021.89}, annote = {Keywords: Quantum Computing, ZH-Calculus, Decision Diagrams} }

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics.
We address the question of the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics.
We introduce a new construction, the discard construction, which transforms any dagger-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal.
Using this construction we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However this construction fails for some fringe cases like the Clifford+T quantum mechanics, as the category does not have enough isometries.

Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Completeness of Graphical Languages for Mixed States Quantum Mechanics (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 108:1-108:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carette_et_al:LIPIcs.ICALP.2019.108, author = {Carette, Titouan and Jeandel, Emmanuel and Perdrix, Simon and Vilmart, Renaud}, title = {{Completeness of Graphical Languages for Mixed States Quantum Mechanics}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {108:1--108:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.108}, URN = {urn:nbn:de:0030-drops-106844}, doi = {10.4230/LIPIcs.ICALP.2019.108}, annote = {Keywords: Quantum Computing, Quantum Categorical Mechanics, Category Theory, Mixed States, Completely Positive Maps} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language - i.e. the ability to derive any true equation - is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles.
We introduce a generalised supplementarity - called cyclotomic supplementarity - which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.
We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart, and Quanlong Wang. ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{jeandel_et_al:LIPIcs.MFCS.2017.11, author = {Jeandel, Emmanuel and Perdrix, Simon and Vilmart, Renaud and Wang, Quanlong}, title = {{ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {11:1--11:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.11}, URN = {urn:nbn:de:0030-drops-81173}, doi = {10.4230/LIPIcs.MFCS.2017.11}, annote = {Keywords: Categorical Quantum Mechanincs, ZX-Calculus, Completeness, Cyclotomic Supplmentarity, Clifford+T} }

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