9 Search Results for "van de Wetering, John"

Document
DOI: 10.4230/DagRep.15.9.58
Quantum Error Correction Meets ZX-Calculus (Dagstuhl Seminar 25382)

Authors: Miriam Backens, Aleks Kissinger, John van de Wetering, Michael Vasmer, and Sarah Meng Li

Published in: Dagstuhl Reports, Volume 15, Issue 9 (2026)

Abstract
This report documents the Dagstuhl Seminar 25382 "Quantum Error Correction Meets ZX-Calculus". The report consists of an executive summary, as well as abstracts on talks, working groups, panel discussions, and open problems.
Cite as

Miriam Backens, Aleks Kissinger, John van de Wetering, Michael Vasmer, and Sarah Meng Li. Quantum Error Correction Meets ZX-Calculus (Dagstuhl Seminar 25382). In Dagstuhl Reports, Volume 15, Issue 9, pp. 58-70, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@Article{backens_et_al:DagRep.15.9.58,
  author =	{Backens, Miriam and Kissinger, Aleks and van de Wetering, John and Vasmer, Michael and Li, Sarah Meng},
  title =	{{Quantum Error Correction Meets ZX-Calculus (Dagstuhl Seminar 25382)}},
  pages =	{58--70},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2026},
  volume =	{15},
  number =	{9},
  editor =	{Backens, Miriam and Kissinger, Aleks and van de Wetering, John and Vasmer, Michael and Li, Sarah Meng},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.15.9.58},
  URN =		{urn:nbn:de:0030-drops-249787},
  doi =		{10.4230/DagRep.15.9.58},
  annote =	{Keywords: fault-tolerance, quantum error correction, ZX-calculus}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.85
Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits

Authors: Neil J. Ross and Scott Wesley

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Many promising quantum algorithms in economics, medical science, and material science rely on circuits that are parameterized by a large number of angles. To ensure that these algorithms are efficient, these parameterized circuits must be heavily optimized. However, most quantum circuit optimizers are not verified, so this procedure is known to be error-prone. For this reason, there is growing interest in the design of equivalence checking algorithms for parameterized quantum circuits. In this paper, we define a generalized class of parameterized circuits with arbitrary rotations and show that this problem is decidable for cyclotomic gate sets. We propose a cutoff-based procedure which reduces the problem of verifying the equivalence of parameterized quantum circuits to the problem of verifying the equivalence of finitely many parameter-free quantum circuits. Because the number of parameter-free circuits grows exponentially with the number of parameters, we also propose a probabilistic variant of the algorithm for cases when the number of parameters is intractably large. We show that our techniques extend to equivalence modulo global phase, and describe an efficient angle sampling procedure for cyclotomic gate sets.
Cite as

Neil J. Ross and Scott Wesley. Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ross_et_al:LIPIcs.MFCS.2025.85,
  author =	{Ross, Neil J. and Wesley, Scott},
  title =	{{Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{85:1--85:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.85},
  URN =		{urn:nbn:de:0030-drops-241921},
  doi =		{10.4230/LIPIcs.MFCS.2025.85},
  annote =	{Keywords: Quantum Circuits, Parameterized Equivalence Checking}
}
Document
DOI: 10.4230/LIPIcs.CP.2025.38
Reducing Quantum Circuit Synthesis to #SAT

Authors: Dekel Zak, Jingyi Mei, Jean-Marie Lagniez, and Alfons Laarman

Published in: LIPIcs, Volume 340, 31st International Conference on Principles and Practice of Constraint Programming (CP 2025)

Abstract
Quantum circuit synthesis is the task of decomposing a given quantum operator into a sequence of elementary quantum gates. Since the finite target gate set cannot exactly implement any given operator, approximation is often necessary. Model counting, or #SAT, has recently been demonstrated as a promising new approach for tackling core problems in quantum circuit analysis. In this work, we show for the first time that the universal quantum circuit synthesis problem can be reduced to maximum model counting. We formulate a #SAT encoding for exact and approximate depth-optimal quantum circuit synthesis into the Clifford+T gate set. We evaluate our method with an open-source implementation that uses the maximum model counter d4Max as a backend. For this purpose, we extended d4Max with support for complex and negative weights to represent amplitudes. Experimental results show that existing classical tools have potential for the quantum circuit synthesis problem.
Cite as

Dekel Zak, Jingyi Mei, Jean-Marie Lagniez, and Alfons Laarman. Reducing Quantum Circuit Synthesis to #SAT. In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 38:1-38:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{zak_et_al:LIPIcs.CP.2025.38,
  author =	{Zak, Dekel and Mei, Jingyi and Lagniez, Jean-Marie and Laarman, Alfons},
  title =	{{Reducing Quantum Circuit Synthesis to #SAT}},
  booktitle =	{31st International Conference on Principles and Practice of Constraint Programming (CP 2025)},
  pages =	{38:1--38:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-380-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{340},
  editor =	{de la Banda, Maria Garcia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2025.38},
  URN =		{urn:nbn:de:0030-drops-238997},
  doi =		{10.4230/LIPIcs.CP.2025.38},
  annote =	{Keywords: Maximum weighted model counting, quantum circuit synthesis}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.22
Branch Sequentialization in Quantum Polytime

Authors: Emmanuel Hainry, Romain Péchoux, and Mário Silva

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
Quantum algorithms leverage the use of quantumly-controlled data in order to achieve computational advantage. This implies that the programs use constructs depending on quantum data and not just classical data such as measurement outcomes. Current compilation strategies for quantum control flow involve compiling the branches of a quantum conditional, either in-depth or in-width, which in general leads to circuits of exponential size. This problem is coined as the branch sequentialization problem. We introduce and study a compilation technique for avoiding branch sequentialization on a language that is sound and complete for quantum polynomial time, thus, improving on existing polynomial-size-preserving compilation techniques.
Cite as

Emmanuel Hainry, Romain Péchoux, and Mário Silva. Branch Sequentialization in Quantum Polytime. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 22:1-22:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hainry_et_al:LIPIcs.FSCD.2025.22,
  author =	{Hainry, Emmanuel and P\'{e}choux, Romain and Silva, M\'{a}rio},
  title =	{{Branch Sequentialization in Quantum Polytime}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{22:1--22:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.22},
  URN =		{urn:nbn:de:0030-drops-236373},
  doi =		{10.4230/LIPIcs.FSCD.2025.22},
  annote =	{Keywords: Quantum Programs, Implicit Computational Complexity, Quantum Circuits}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.20
Simple Qudit ZX and ZH Calculi, via Integrals

Authors: Niel de Beaudrap and Richard D. P. East

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
The ZX calculus and ZH calculus use diagrams to denote and compute properties of quantum operations, using "rewrite rules" to transform between diagrams which denote the same operator through a functorial semantic map. Different semantic maps give rise to different rewrite systems, which may prove more convenient for different purposes. Using discrete measures, we describe semantic maps for ZX and ZH diagrams, well-suited to analyse unitary circuits and measurements on qudits of any fixed dimension D > 1 as a single "ZXH-calculus". We demonstrate rewrite rules for the "stabiliser fragment" of the ZX calculus and a "multicharacter fragment" of the ZH calculus.
Cite as

Niel de Beaudrap and Richard D. P. East. Simple Qudit ZX and ZH Calculi, via Integrals. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{debeaudrap_et_al:LIPIcs.MFCS.2024.20,
  author =	{de Beaudrap, Niel and East, Richard D. P.},
  title =	{{Simple Qudit ZX and ZH Calculi, via Integrals}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.20},
  URN =		{urn:nbn:de:0030-drops-205761},
  doi =		{10.4230/LIPIcs.MFCS.2024.20},
  annote =	{Keywords: ZX-calculus, ZH-calculus, qudits, string diagrams, discrete integrals}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.5
Classical Simulation of Quantum Circuits with Partial and Graphical Stabiliser Decompositions

Authors: Aleks Kissinger, John van de Wetering, and Renaud Vilmart

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

Abstract
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.
Cite as

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
DOI: 10.4230/LIPIcs.TQC.2022.12
Qutrit Metaplectic Gates Are a Subset of Clifford+T

Authors: Andrew N. Glaudell, Neil J. Ross, John van de Wetering, and Lia Yeh

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

Abstract
A popular universal gate set for quantum computing with qubits is Clifford+T, as this can be readily implemented on many fault-tolerant architectures. For qutrits, there is an equivalent T gate, that, like its qubit analogue, makes Clifford+T approximately universal, is injectable by a magic state, and supports magic state distillation. However, it was claimed that a better gate set for qutrits might be Clifford+R, where R = diag(1,1,-1) is the metaplectic gate, as certain protocols and gates could more easily be implemented using the R gate than the T gate. In this paper we show that the qutrit Clifford+R unitaries form a strict subset of the Clifford+T unitaries when we have at least two qutrits. We do this by finding a direct decomposition of R ⊗ 𝕀 as a Clifford+T circuit and proving that the T gate cannot be exactly synthesized in Clifford+R. This shows that in fact the T gate is more expressive than the R gate. Moreover, we additionally show that it is impossible to find a single-qutrit Clifford+T decomposition of the R gate, making our result tight.
Cite as

Andrew N. Glaudell, Neil J. Ross, John van de Wetering, and Lia Yeh. Qutrit Metaplectic Gates Are a Subset of Clifford+T. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{glaudell_et_al:LIPIcs.TQC.2022.12,
  author =	{Glaudell, Andrew N. and Ross, Neil J. and van de Wetering, John and Yeh, Lia},
  title =	{{Qutrit Metaplectic Gates Are a Subset of Clifford+T}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{12:1--12:15},
  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.2022.12},
  URN =		{urn:nbn:de:0030-drops-165195},
  doi =		{10.4230/LIPIcs.TQC.2022.12},
  annote =	{Keywords: Quantum computation, qutrits, gate synthesis, metaplectic gate, Clifford+T}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2022.119
Circuit Extraction for ZX-Diagrams Can Be #P-Hard

Authors: Niel de Beaudrap, Aleks Kissinger, and John van de Wetering

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
The ZX-calculus is a graphical language for reasoning about quantum computation using ZX-diagrams, a certain flexible generalisation of quantum circuits that can be used to represent linear maps from m to n qubits for any m,n ≥ 0. Some applications for the ZX-calculus, such as quantum circuit optimisation and synthesis, rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. While several sufficient conditions are known for describing families of ZX-diagrams that can be efficiently transformed back into circuits, it has previously been conjectured that the general problem of circuit extraction is hard. That is, that it should not be possible to efficiently convert an arbitrary ZX-diagram describing a unitary linear map into an equivalent quantum circuit. In this paper we prove this conjecture by showing that the circuit extraction problem is #P-hard, and so is itself at least as hard as strong simulation of quantum circuits. In addition to our main hardness result, which relies specifically on the circuit representation, we give a representation-agnostic hardness result. Namely, we show that any oracle that takes as input a ZX-diagram description of a unitary and produces samples of the output of the associated quantum computation enables efficient probabilistic solutions to NP-complete problems.
Cite as

Niel de Beaudrap, Aleks Kissinger, and John van de Wetering. Circuit Extraction for ZX-Diagrams Can Be #P-Hard. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 119:1-119:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{debeaudrap_et_al:LIPIcs.ICALP.2022.119,
  author =	{de Beaudrap, Niel and Kissinger, Aleks and van de Wetering, John},
  title =	{{Circuit Extraction for ZX-Diagrams Can Be #P-Hard}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{119:1--119:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.119},
  URN =		{urn:nbn:de:0030-drops-164601},
  doi =		{10.4230/LIPIcs.ICALP.2022.119},
  annote =	{Keywords: ZX-calculus, circuit extraction, quantum circuits, #P}
}
Document
DOI: 10.4230/LIPIcs.TQC.2020.11
Fast and Effective Techniques for T-Count Reduction via Spider Nest Identities

Authors: Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang

Published in: LIPIcs, Volume 158, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

Abstract
In fault-tolerant quantum computing systems, realising (approximately) universal quantum computation is usually described in terms of realising Clifford+T operations, which is to say a circuit of CNOT, Hadamard, and π/2-phase rotations, together with T operations (π/4-phase rotations). For many error correcting codes, fault-tolerant realisations of Clifford operations are significantly less resource-intensive than those of T gates, which motivates finding ways to realise the same transformation involving T-count (the number of T gates involved) which is as low as possible. Investigations into this problem [Matthew Amy et al., 2013; Gosset et al., 2014; Matthew Amy et al., 2014; Matthew Amy et al., 2018; Earl T. Campbell and Mark Howard, 2017; Matthew Amy and Michele Mosca, 2019] has led to observations that this problem is closely related to NP-hard tensor decomposition problems [Luke E. Heyfron and Earl T. Campbell, 2018] and is tantamount to the difficult problem of decoding exponentially long Reed-Muller codes [Matthew Amy and Michele Mosca, 2019]. This problem then presents itself as one for which must be content in practise with approximate optimisation, in which one develops an array of tactics to be deployed through some pragmatic strategy. In this vein, we describe techniques to reduce the T-count, based on the effective application of "spider nest identities": easily recognised products of parity-phase operations which are equivalent to the identity operation. We demonstrate the effectiveness of such techniques by obtaining improvements in the T-counts of a number of circuits, in run-times which are typically less than the time required to make a fresh cup of coffee.
Cite as

Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang. Fast and Effective Techniques for T-Count Reduction via Spider Nest Identities. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{debeaudrap_et_al:LIPIcs.TQC.2020.11,
  author =	{de Beaudrap, Niel and Bian, Xiaoning and Wang, Quanlong},
  title =	{{Fast and Effective Techniques for T-Count Reduction via Spider Nest Identities}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{11:1--11:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Flammia, Steven T.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.11},
  URN =		{urn:nbn:de:0030-drops-120705},
  doi =		{10.4230/LIPIcs.TQC.2020.11},
  annote =	{Keywords: T-count, Parity-phase operations, Phase gadgets, Clifford hierarchy, ZX calculus}
}
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