Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

Ross Duncan; Aleks Kissinger; Simon Perdrix; John van de Wetering

doi:10.22331/q-2020-06-04-279

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

Ross Duncan^1,2, Aleks Kissinger³, Simon Perdrix⁴, and John van de Wetering⁵

¹University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
²Cambridge Quantum Computing Ltd, 9a Bridge Street, Cambridge CB2 1UB, UK
³Department of Computer Science, University of Oxford
⁴CNRS LORIA, Inria-MOCQUA, Université de Lorraine, F 54000 Nancy, France
⁵Institute for Computing and Information Sciences, Radboud University Nijmegen

Published:	2020-06-04, volume 4, page 279
Eprint:	arXiv:1902.03178v6
Doi:	https://doi.org/10.22331/q-2020-06-04-279
Citation:	Quantum 4, 279 (2020).

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Abstract

We present a completely new approach to quantum circuit optimisation, based on the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we give a simplification strategy for ZX-diagrams based on the two graph transformations of local complementation and pivoting and show that the resulting reduced diagram can be transformed back into a quantum circuit. While little is known about extracting circuits from arbitrary ZX-diagrams, we show that the underlying graph of our simplified ZX-diagram always has a graph-theoretic property called generalised flow, which in turn yields a deterministic circuit extraction procedure. For Clifford circuits, this extraction procedure yields a new normal form that is both asymptotically optimal in size and gives a new, smaller upper bound on gate depth for nearest-neighbour architectures. For Clifford+T and more general circuits, our technique enables us to to `see around' gates that obstruct the Clifford structure and produce smaller circuits than naïve `cut-and-resynthesise' methods.

Popular summary

Quantum circuits are a de facto assembly language for quantum software. Programs are described as list of primitive operations, or gates, which are run in sequence on a quantum computer to perform a computation. Just like with classical software, there is more that one way to write a program to do the same job, and so it's important to find programs that do that job as quickly and cheaply as possible. Looking at quantum circuits just as lists of gates doesn't tell us a whole lot about what computation is being performed, or how it might be optimised. However, if we "break open" quantum gates, we see a rich graphical/algebraic structure inside called the ZX-calculus.

In this paper, we give a technique for optimising quantum circuits that first breaks the gates open to reveal a graph-like structure underneath. We then give a strategy for reducing these graphs in size without changing the computation they represent, then extracting a new, smaller circuit out of the result.

We have implemented the technique described in this paper in a Python tool called PyZX:
https://github.com/Quantomatic/pyzx

For a 2-minute intro to PyZX, see this YouTube video: https://www.youtube.com/watch?v=iC-KVdB8pf0

If that leaves you wanting more, there is also a 40-minute talk about this and more on YouTube: https://www.youtube.com/watch?v=JafI_LZts2g

► BibTeX data

@article{Duncan2020graphtheoretic,
  doi = {10.22331/q-2020-06-04-279},
  url = {https://doi.org/10.22331/q-2020-06-04-279},
  title = {Graph-theoretic {S}implification of {Q}uantum {C}ircuits with the {ZX}-calculus},
  author = {Duncan, Ross and Kissinger, Aleks and Perdrix, Simon and van de Wetering, John},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {279},
  month = jun,
  year = {2020}
}

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[64] Giovanni de Felice and Bob Coecke, "Quantum Linear Optics via String Diagrams", Electronic Proceedings in Theoretical Computer Science 394, 83 (2023).

[65] Tristan Cam, Cyril Gavoille, Yvan Le Borgne, and Simon Martiel, Lecture Notes in Computer Science 15716, 54 (2025) ISBN:978-3-031-97062-7.

[66] Ludwig Schmid, Korbinian Staudacher, and Robert Wille, 2026 Design, Automation & Test in Europe Conference (DATE) 1 (2026) ISBN:978-3-9826741-1-7.

[67] Timothee Goubault de Brugiere, Marc Baboulin, Benoit Valiron, Simon Martiel, and Cyril Allouche, "Reducing the Depth of Linear Reversible Quantum Circuits", IEEE Transactions on Quantum Engineering 2, 1 (2021).

[68] Einat Buznach Ahituv, Debanjan Chowdhury, and Jonathan Ruhman, "Unveiling a Hidden Percolation Transition in Monitored Clifford Circuits: Inroads from ZX Calculus", Physical Review Letters 135 5, 050402 (2025).

[69] John van de Wetering, Richie Yeung, Tuomas Laakkonen, and Aleks Kissinger, "Optimal compilation of parametrised quantum circuits", Quantum 9, 1828 (2025).

[70] Pascal Baßler, Matthias Zipper, Christopher Cedzich, Markus Heinrich, Patrick H. Huber, Michael Johanning, and Martin Kliesch, "Synthesis of and compilation with time-optimal multi-qubit gates", Quantum 7, 984 (2023).

[71] Matthew Sutcliffe and Aleks Kissinger, "Procedurally Optimised ZX-Diagram Cutting for Efficient T-Decomposition in Classical Simulation", Electronic Proceedings in Theoretical Computer Science 406, 63 (2024).

[72] Fanxu Meng, Yuxiang Liu, Lu Wang, Weiwei Zhou, and Xiangzhen Zhou, "Low-depth quantum approximate optimization algorithm for maximum likelihood detection in massive MIMO", Quantum Information Processing 24 9, 294 (2025).

[73] Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron, 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 1 (2023) ISBN:979-8-3503-3587-3.

[74] Robert I. Booth and Damian Markham, "Flow conditions for continuous variable measurement-based quantum computing", Quantum 7, 1146 (2023).

[75] Li Zhou, Gilles Barthe, Pierre-Yves Strub, Junyi Liu, and Mingsheng Ying, "CoqQ: Foundational Verification of Quantum Programs", Proceedings of the ACM on Programming Languages 7 POPL, 833 (2023).

[76] Daniele Cuomo, Springer Theses 61 (2024) ISBN:978-3-031-73807-4.

[77] Keyu Xiong, Tao Shang, Chenyi Zhang, and Yuchen Liu, "ZX-Diagram T-Count Reduction Based on Proximal Policy Optimization", Intelligent Computing 5, 0418 (2026).

[78] Matthew Sutcliffe and Aleks Kissinger, "Fast Classical Simulation of Quantum Circuits via Parametric Rewriting in the ZX-Calculus", Electronic Proceedings in Theoretical Computer Science 426, 247 (2025).

[79] Lukas Burgholzer and Robert Wille, Design Automation Tools and Software for Quantum Computing 81 (2026) ISBN:978-3-032-06769-2.

[80] Mingsheng Ying, Foundations of Quantum Programming 367 (2024) ISBN:9780443159428.

[81] Niel de Beaudrap, "Well-tempered ZX and ZH Calculi", Electronic Proceedings in Theoretical Computer Science 340, 13 (2021).

[82] Tobias Stollenwerk and Stuart Hadfield, "Diagrammatic Analysis for Parameterized Quantum Circuits", Electronic Proceedings in Theoretical Computer Science 394, 262 (2023).

[83] Jingyi Mei, Jan Martens, and Alfons Laarman, Lecture Notes in Computer Science 15373, 17 (2025) ISBN:978-3-031-77018-0.

[84] Robert Wille, Lucas Berent, Tobias Forster, Jagatheesan Kunasaikaran, Kevin Mato, Tom Peham, Nils Quetschlich, Damian Rovara, Aaron Sander, Ludwig Schmid, Daniel Schönberger, Yannick Stade, and Lukas Burgholzer, 2024 IEEE International Conference on Quantum Software (QSW) 1 (2024) ISBN:979-8-3503-6847-5.

[85] Jordi Riu, Jan Nogué, Gerard Vilaplana, Artur Garcia-Saez, and Marta P. Estarellas, "Reinforcement Learning Based Quantum Circuit Optimization via ZX-Calculus", Quantum 9, 1758 (2025).

[86] Alexis Toumi, Richie Yeung, and Giovanni de Felice, "Diagrammatic Differentiation for Quantum Machine Learning", Electronic Proceedings in Theoretical Computer Science 343, 132 (2021).

[87] Bob Coecke, Aleks Kissinger, Stefano Gogioso, Selma Dündar-Coecke, Caterina Puca, Lia Yeh, Muhammad Hamza Waseem, Emmanuel M. Pothos, Sieglinde Pfaendler, Vincent Wang-Mascianica, Thomas Cervoni, Ferdi Tomassini, Vincent Anandraj, Peter Sigrist, and Ilyas Khan, "High schoolers excel at Oxford quantum course using pictorial mathematics", Journal of Physics: Conference Series 3189 1, 012008 (2026).

[88] Priyanka Mukhopadhyay, Nathan Wiebe, and Hong Tao Zhang, "Synthesizing efficient circuits for Hamiltonian simulation", npj Quantum Information 9 1, 31 (2023).

[89] Wonho Jang, Koji Terashi, Masahiko Saito, Christian W. Bauer, Benjamin Nachman, Yutaro Iiyama, Tomoe Kishimoto, Ryunosuke Okubo, Ryu Sawada, Junichi Tanaka, C. Biscarat, S. Campana, B. Hegner, S. Roiser, C.I. Rovelli, and G.A. Stewart, "Quantum Gate Pattern Recognition and Circuit Optimization for Scientific Applications", EPJ Web of Conferences 251, 03023 (2021).

[90] Felix Günther Gemeinhardt, Stefan Klikovits, and Manuel Wimmer, "Gequpi: Quantum Program Improvement with Multi-Objective Genetic Programming", (2023).

[91] Liam Hurwitz, Kamalika Datta, Abhoy Kole, and Rolf Drechsler, Lecture Notes in Computer Science 14680, 201 (2024) ISBN:978-3-031-62075-1.

[92] Maximilian Nägele and Florian Marquardt, "Optimizing ZX-diagrams with deep reinforcement learning", Machine Learning: Science and Technology 5 3, 035077 (2024).

[93] Yusei Mori, Hideaki Hakoshima, and Keisuke Fujii, "Nontrivial Multiproduct Commutation Relation Toward Reducing T -Count in Sequential Pauli-Based Computation", PRX Quantum 7 2, 020345 (2026).

[94] Alexandru Paler, Lucian Sasu, Adrian-Cătălin Florea, and Răzvan Andonie, "Machine Learning Optimization of Quantum Circuit Layouts", ACM Transactions on Quantum Computing 4 2, 1 (2023).

[95] Andrew Vlasic and Anh Pham, "Scoring Anomalous Vertices Through Quantum Walks", Annalen der Physik 537 5, 2400282 (2025).

[96] Pedro M. Q. Cruz and Bruno Murta, "Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging", APL Quantum 1 1, 016105 (2024).

[97] Agustín Borgna, Simon Perdrix, and Benoît Valiron, Lecture Notes in Computer Science 13008, 121 (2021) ISBN:978-3-030-89050-6.

[98] Xin Hong, Chenjian Li, Aochu Dai, Runhong He, and Shenggang Ying, 2026 Design, Automation & Test in Europe Conference (DATE) 1 (2026) ISBN:978-3-9826741-1-7.

[99] Titouan Carette, Yohann D'Anello, and Simon Perdrix, "Quantum Algorithms and Oracles with the Scalable ZX-calculus", Electronic Proceedings in Theoretical Computer Science 343, 193 (2021).

[100] Aravind Joshi, Akshara Kairali, Renju Raju, Adithya Athreya, and Reena Monica P, "Quantum Circuit Optimization of Arithmetic Circuits using ZX Calculus", International Journal of Innovative Technology and Exploring Engineering 13 2, 26 (2024).

[101] John van de Wetering, "Constructing quantum circuits with global gates", New Journal of Physics 23 4, 043015 (2021).

[102] Hao Tang, Yu Guo, Weixiao Sun, Xiao-Min Hu, Bi-Heng Liu, Zhaohui Wei, Yun-Feng Huang, Yong-Jian Han, Chuan-Feng Li, and Guang-Can Guo, "Experimental Equivalence Checking of Quantum Circuits by Nonlocality", Physical Review Letters 133 26, 260602 (2024).

[103] Alejandro Villoria, Henning Basold, and Alfons Laarman, "Optimisation and synthesis of quantum circuits with global gates", Quantum Science and Technology 11 1, 015040 (2026).

[104] Mingsheng Ying, Li Zhou, and Gilles Barthe, "Laws of Quantum Programming", ACM Transactions on Software Engineering and Methodology 35 7, 1 (2026).

[105] Kevin Mato, Martin Ringbauer, Stefan Hillmich, and Robert Wille, Proceedings of the 28th Asia and South Pacific Design Automation Conference 202 (2023) ISBN:9781450397834.

[106] Ion Nechita, Clément Pellegrini, and Denis Rochette, "The asymmetric quantum cloning region", Letters in Mathematical Physics 113 3, 74 (2023).

[107] Robert I Booth, Aleks Kissinger, Damian Markham, Clément Meignant, and Simon Perdrix, "Outcome determinism in measurement-based quantum computation with qudits", Journal of Physics A: Mathematical and Theoretical 56 11, 115303 (2023).

[108] Armando Bellante, Tommaso Fioravanti, Michele Carminati, Stefano Zanero, and Alessandro Luongo, "Evaluating the potential of quantum machine learning in cybersecurity: A case-study on PCA-based intrusion detection systems", Computers & Security 154, 104341 (2025).

[109] Dimiter Ostrev, "Canonical Form and Finite Blocklength Bounds for Stabilizer Codes", IEEE Transactions on Information Theory 72 6, 3979 (2026).

[110] Richard D.P. East, John van de Wetering, Nicholas Chancellor, and Adolfo G. Grushin, "AKLT-States as ZX-Diagrams: Diagrammatic Reasoning for Quantum States", PRX Quantum 3 1, 010302 (2022).

[111] Hyunjoon Shin, Seokhyeon Lee, Myeongjin Kwak, and Yongtae Kim, Proceedings of the 40th ACM International Conference on Supercomputing 1296 (2026) ISBN:9798400725227.

[112] Aaron Sander, Lukas Burgholzer, and Robert Wille, "Equivalence checking of quantum circuits via intermediary matrix product operator", Physical Review Research 7 2, 023261 (2025).

[113] Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "Completeness of Graphical Languages for Mixed State Quantum Mechanics", ACM Transactions on Quantum Computing 2 4, 1 (2021).

[114] Bing Li, Yangchao Yu, Fubo Ma, ZiYang Dong, and Le Zhang, "Developing a quantum computing model for sequence annotation of interferon protein", Computational and Structural Biotechnology Journal 30, 134 (2025).

[115] Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, and Alexis Toumi, Quantum Computing in the Arts and Humanities 277 (2022) ISBN:978-3-030-95537-3.

[116] Nathan Claudet and Simon Perdrix, Lecture Notes in Computer Science 14760, 136 (2025) ISBN:978-3-031-75408-1.

[117] Korbinian Staudacher, Ludwig Schmid, Johannes Zeiher, Robert Wille, and Dieter Kranzlmüller, "Multi-controlled Phase Gate Synthesis with ZX-calculus applied to Neutral Atom Hardware", Electronic Proceedings in Theoretical Computer Science 406, 96 (2024).

[118] M. Schwetz and R. M. Noack, "Three-qubit Deutsch–Jozsa in measurement-based quantum computing", International Journal of Quantum Information 22 07, 2350046 (2024).

[119] Felix Gemeinhardt, Stefan Klikovits, and Manuel Wimmer, "GeQuPI: Quantum Program Improvement with Multi-Objective Genetic Programming", Journal of Systems and Software 219, 112223 (2025).

[120] Lina Vandré, Boxuan Jing, Yu Xiang, Otfried Gühne, and Qiongyi He, "Graphical Framework for Non-Gaussian Quantum States", Quantum 9, 1809 (2025).

[121] Bruno Murta, Pedro M. Q. Cruz, and J. Fernández-Rossier, "Preparing valence-bond-solid states on noisy intermediate-scale quantum computers", Physical Review Research 5 1, 013190 (2023).

[122] Johannes Nokkala, Jyrki Piilo, and Ginestra Bianconi, "Complex quantum networks: a topical review", Journal of Physics A: Mathematical and Theoretical 57 23, 233001 (2024).

[123] Nawel Huenchuleo, Samuel Sepúlveda, and Alejandro Fernández, "QC-MM: A Metadata and Schema Model for Traceable Quantum-Circuit Experiments", Applied Sciences 16 13, 6346 (2026).

[124] Tommy McElvanney and Miriam Backens, "Complete Flow-Preserving Rewrite Rules for MBQC Patterns with Pauli Measurements", Electronic Proceedings in Theoretical Computer Science 394, 66 (2023).

[125] Bob Coecke, Dominic Horsman, Aleks Kissinger, and Quanlong Wang, "Kindergarden quantum mechanics graduates ...or how I learned to stop gluing LEGO together and love the ZX-calculus", Theoretical Computer Science 897, 1 (2022).

[126] Xiaofeng Gao, Zhijin Guan, Shiguang Feng, and Yibo Jiang, "Quantum Circuit Template Matching Optimization Method for Constrained Connectivity", Axioms 12 7, 687 (2023).

[127] Bob Coecke, The Quantum-Like Revolution 247 (2023) ISBN:978-3-031-12985-8.

[128] Zipeng Wu, Song Cheng, and Bei Zeng, "Efficient construction of stabilizer codes via ZX-calculus", Academia Quantum 3 2(2026).

[129] Santiago Núñcz-Corrales, Marcos Frenkel, and Bruno Abreu, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1001 (2023) ISBN:979-8-3503-4323-6.

[130] Philipp Schmidt, Sören Arlt, Carlos Ruiz-Gonzalez, Xuemei Gu, Carla Rodríguez, and Mario Krenn, "Virtual reality for understanding artificial-intelligence-driven scientific discovery with an application in quantum optics", Machine Learning: Science and Technology 5 3, 035045 (2024).

[131] Narayanan Rengaswamy, Robert Calderbank, Swanand Kadhe, and Henry D. Pfister, "Logical Clifford Synthesis for Stabilizer Codes", IEEE Transactions on Quantum Engineering 1, 1 (2020).

[132] David Winderl, Qunsheng Huang, and Christian B. Mendl, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 837 (2023) ISBN:979-8-3503-4323-6.

[133] Thierry N Kaldenbach and Matthias Heller, "Mapping quantum circuits to shallow-depth measurement patterns based on graph states", Quantum Science and Technology 10 1, 015010 (2025).

[134] Jiawen Li, Dongdong Li, Ziyang Wang, He Sun, Hongzhen Chen, Tianxiang Cui, and Yujian Ye, "StateQ: A novel framework for intuitive and accessible quantum programming", Quantum Machine Intelligence 8 1, 60 (2026).

[135] Tommy McElvanney and Miriam Backens, "Flow-preserving ZX-calculus Rewrite Rules for Optimisation and Obfuscation", Electronic Proceedings in Theoretical Computer Science 384, 203 (2023).

[136] Boldizsár Poór, Razin A. Shaikh, and Quanlong Wang, "ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces", Electronic Proceedings in Theoretical Computer Science 426, 127 (2025).

[137] Korbinian Staudacher, Tobias Guggemos, Sophia Grundner-Culemann, and Wolfgang Gehrke, "Reducing 2-QuBit Gate Count for ZX-Calculus based Quantum Circuit Optimization", Electronic Proceedings in Theoretical Computer Science 394, 29 (2023).

[138] Mariia Makarova, Davide Pastorello, Valter Cavecchia, Enrico Blanzieri, Kamil Khadiev, and Alexander Rumyantsev, "Quantum Circuit Optimization Techniques: A Literature Survey", Quantum Information & Computation 26 1, 154 (2026).

[139] Lukas Burgholzer, Yannick Stade, Tom Peham, and Robert Wille, "MQT Core: The Backbone of the Munich Quantum Toolkit (MQT)", Journal of Open Source Software 10 108, 7478 (2025).

[140] Aleks Kissinger and John van de Wetering, "Reducing the number of non-Clifford gates in quantum circuits", Physical Review A 102 2, 022406 (2020).

[141] Titouan Carette, Marc de Visme, and Simon Perdrix, 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 1 (2021) ISBN:978-1-6654-4895-6.

[142] Sergey Bravyi and Dmitri Maslov, "Hadamard-Free Circuits Expose the Structure of the Clifford Group", IEEE Transactions on Information Theory 67 7, 4546 (2021).

[143] Tom Ewen, Ivica Turkalj, Patrick Holzer, and Mark-Oliver Wolf, "Application of ZX-calculus to quantum architecture search", Quantum Machine Intelligence 7 1, 34 (2025).

[144] Selma Dündar-Coecke, Lia Yeh, Caterina Puca, Sieglinde M.-L. Pfaendler, Muhammad Hamza Waseem, Thomas Cervoni, Aleks Kissinger, Stefano Gogioso, and Bob Coecke, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 21 (2023) ISBN:979-8-3503-4323-6.

[145] Lu Li, Kaifeng Bu, Dax Enshan Koh, Arthur Jaffe, and Seth Lloyd, "Wasserstein complexity of quantum circuits", Journal of Physics A: Mathematical and Theoretical 58 26, 265302 (2025).

[146] José L. Gómez-Sirvent, Antonio Fernández-Caballero, and Paulo Novais, "Functional near-infrared spectroscopy for the detection of fear using parameterized quantum circuits", Scientific Reports 15 1, 44911 (2025).

[147] Piotr Mitosek, "Pauli Flow on Open Graphs with Unknown Measurement Labels", Electronic Proceedings in Theoretical Computer Science 406, 117 (2024).

[148] Timothée Goubault de Brugière and Simon Martiel, "Shallower CNOT Circuits on Realistic Quantum Hardware", ACM Transactions on Quantum Computing 6 2, 1 (2025).

[149] Waldemir Cambiucci, Regina Melo Silveira, and Wilson Vicente Ruggiero, "Co-design threading model and circuit cutting for static and adaptive quantum circuits", Quantum Information Processing 25 4, 118 (2026).

[150] Hironari Nagayoshi, Warit Asavanant, Ryuhoh Ide, Kosuke Fukui, Atsushi Sakaguchi, Jun-ichi Yoshikawa, Nicolas C. Menicucci, and Akira Furusawa, "ZX graphical calculus for continuous-variable quantum processes", Physical Review Research 7 3, 033141 (2025).

[151] Razin A. Shaikh, Quanlong Wang, and Richie Yeung, "How to Sum and Exponentiate Hamiltonians in ZXW Calculus", Electronic Proceedings in Theoretical Computer Science 394, 236 (2023).

[152] Lukas Burgholzer and Robert Wille, Design Automation Tools and Software for Quantum Computing 163 (2026) ISBN:978-3-032-06769-2.

[153] Chris N. Self, Sofyan Iblisdir, Gavin K. Brennen, and Konstantinos Meichanetzidis, "Estimating the Jones polynomial for Ising anyons on noisy quantum computers", Physical Review Research 8 2, 023168 (2026).

[154] Guedong Park, Yong Siah Teo, and Hyunseok Jeong, "Resource-efficient shadow tomography using equatorial stabilizer measurements", Physical Review Research 7 3, 033097 (2025).

[155] Antonio D. Corcoles, Abhinav Kandala, Ali Javadi-Abhari, Douglas T. McClure, Andrew W. Cross, Kristan Temme, Paul D. Nation, Matthias Steffen, and Jay M. Gambetta, "Challenges and Opportunities of Near-Term Quantum Computing Systems", IEEE Proceedings 108 8, 1338 (2020).

[156] Niel de Beaudrap and Dominic Horsman, "The ZX calculus is a language for surface code lattice surgery", Quantum 4, 218 (2020).

[157] Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, and Alexis Toumi, "Foundations for Near-Term Quantum Natural Language Processing", arXiv:2012.03755, (2020).

[158] Alexander Cowtan, Silas Dilkes, Ross Duncan, Will Simmons, and Seyon Sivarajah, "Phase Gadget Synthesis for Shallow Circuits", arXiv:1906.01734, (2019).

[159] Aleks Kissinger and John van de Wetering, "Reducing T-count with the ZX-calculus", arXiv:1903.10477, (2019).

[160] Aleks Kissinger and John van de Wetering, "PyZX: Large Scale Automated Diagrammatic Reasoning", arXiv:1904.04735, (2019).

[161] Aleks Kissinger and Arianne Meijer-van de Griend, "CNOT circuit extraction for topologically-constrained quantum memories", arXiv:1904.00633, (2019).

[162] Julian Berberich, Tobias Fellner, Robert L. Kosut, and Christian Holm, "Robustness of quantum algorithms: Worst-case fidelity bounds and implications for design", arXiv:2509.08481, (2025).

[163] Tom Peham, Lukas Burgholzer, and Robert Wille, "Equivalence Checking of Quantum Circuits With the ZX-Calculus", IEEE Journal on Emerging and Selected Topics in Circuits and Systems 12 3, 662 (2022).

[164] Jonathan Gorard, Manojna Namuduri, and Xerxes D. Arsiwalla, "ZX-Calculus and Extended Wolfram Model Systems II: Fast Diagrammatic Reasoning with an Application to Quantum Circuit Simplification", arXiv:2103.15820, (2021).

[165] Benjamin Rodatz, Boldizsár Poór, and Aleks Kissinger, "Fault Tolerance by Construction", arXiv:2506.17181, (2025).

[166] Aleks Kissinger and John van de Wetering, "Universal MBQC with generalised parity-phase interactions and Pauli measurements", Quantum 3, 134 (2019).

[167] Lucas Slattery and Bryan K. Clark, "Quantum Circuits For Two-Dimensional Isometric Tensor Networks", arXiv:2108.02792, (2021).

[168] Yuanjie Ren, Kaifeng Bu, and Andreas Bauer, "Graphical Calculus for Fermionic Tensors", arXiv:2508.03976, (2025).

[169] Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang, "Fast and effective techniques for T-count reduction via spider nest identities", arXiv:2004.05164, (2020).

[170] Robert I. Booth, Titouan Carette, and Cole Comfort, "Graphical Symplectic Algebra", arXiv:2401.07914, (2024).

[171] Korbinian Staudacher, Ludwig Schmid, Johannes Zeiher, Robert Wille, and Dieter Kranzlmüller, "Multi-controlled Phase Gate Synthesis with ZX-calculus applied to Neutral Atom Hardware", arXiv:2403.10864, (2024).

[172] Robert Wille, Lucas Berent, Tobias Forster, Jagatheesan Kunasaikaran, Kevin Mato, Tom Peham, Nils Quetschlich, Damian Rovara, Aaron Sander, Ludwig Schmid, Daniel Schönberger, Yannick Stade, and Lukas Burgholzer, "The MQT Handbook: A Summary of Design Automation Tools and Software for Quantum Computing", arXiv:2405.17543, (2024).

[173] Maximilian Nägele, Jan Olle, Thomas Fösel, Remmy Zen, and Florian Marquardt, "Tackling Decision Processes with Non-Cumulative Objectives using Reinforcement Learning", arXiv:2405.13609, (2024).

[174] Filipa C. R. Peres and Ernesto F. Galvão, "Reducing depth and measurement weights in Pauli-based computation", Physical Review A 112 6, 062604 (2025).

[175] Prakash Murali, David C. McKay, Margaret Martonosi, and Ali Javadi-Abhari, "Software Mitigation of Crosstalk on Noisy Intermediate-Scale Quantum Computers", arXiv:2001.02826, (2020).

[176] Mateusz Kupper, Dominic Horsman, Chris Heunen, and Niel de Beaudrap, "String Diagrams for Defect-Based Surface Code Computing", arXiv:2508.14672, (2025).

[177] Andrey Boris Khesin, "Quantum Computing from Graphs", arXiv:2501.17959, (2025).

[178] Boldizsár Poór, Razin A. Shaikh, and Quanlong Wang, "ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces", arXiv:2405.10896, (2024).

[179] Bob Coecke, "Compositionality as we see it, everywhere around us", arXiv:2110.05327, (2021).

[180] Tommy McElvanney and Miriam Backens, "Complete Flow-Preserving Rewrite Rules for MBQC Patterns with Pauli Measurements", arXiv:2205.02009, (2022).

[181] Giovanni de Felice and Bob Coecke, "Quantum Linear Optics via String Diagrams", arXiv:2204.12985, (2022).

[182] Alexander Mattick, Maniraman Periyasamy, Christian Ufrecht, Abhishek Y. Dubey, Christopher Mutschler, Axel Plinge, and Daniel D. Scherer, "Optimizing Quantum Circuits via ZX Diagrams using Reinforcement Learning and Graph Neural Networks", arXiv:2504.03429, (2025).

[183] Razin A. Shaikh, Quanlong Wang, and Richie Yeung, "How to Sum and Exponentiate Hamiltonians in ZXW Calculus", arXiv:2212.04462, (2022).

[184] Niel de Beaudrap, Ross Duncan, Dominic Horsman, and Simon Perdrix, "Pauli Fusion: a Computational Model to Realise Quantum Transformations from ZX Terms", arXiv:1904.12817, (2019).

[185] Stach Kuijpers, John van de Wetering, and Aleks Kissinger, "Graphical Fourier Theory and the Cost of Quantum Addition", arXiv:1904.07551, (2019).

[186] Ewan Murphy and Aleks Kissinger, "Global Synthesis of CNOT Circuits with Holes", arXiv:2308.16496, (2023).

[187] Razin A. Shaikh, Lia Yeh, and Stefano Gogioso, "The Focked-up ZX Calculus: Picturing Continuous-Variable Quantum Computation", arXiv:2406.02905, (2024).

[188] Niel de Beaudrap, "Well-tempered ZX and ZH Calculi", arXiv:2006.02557, (2020).

[189] Titouan Carette, Yohann D'Anello, and Simon Perdrix, "Quantum Algorithms and Oracles with the Scalable ZX-calculus", arXiv:2104.01043, (2021).

[190] Tommy McElvanney and Miriam Backens, "Flow-preserving ZX-calculus Rewrite Rules for Optimisation and Obfuscation", arXiv:2304.08166, (2023).

[191] Tuomas Laakkonen, Konstantinos Meichanetzidis, and John van de Wetering, "A Graphical #SAT Algorithm for Formulae with Small Clause Density", arXiv:2212.08048, (2022).

[192] Tobias Fischbach, Pierre Talbot, and Pascal Bouvry, "Exhaustive Search for Quantum Circuit Optimization using ZX Calculus", arXiv:2503.10983, (2025).

[193] Matthew Sutcliffe and Aleks Kissinger, "Procedurally Optimised ZX-Diagram Cutting for Efficient T-Decomposition in Classical Simulation", arXiv:2403.10964, (2024).

[194] Matthew Sutcliffe and Aleks Kissinger, "Fast Classical Simulation of Quantum Circuits via Parametric Rewriting in the ZX-Calculus", arXiv:2403.06777, (2024).

[195] William Cashman, Giovanni de Felice, and Aleks Kissinger, "Finding trail covers: near-optimal decompositions of graph states as linear fusion networks", arXiv:2508.18375, (2025).

[196] Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "Completeness of Graphical Languages for Mixed States Quantum Mechanics", arXiv:1902.07143, (2019).

[197] Korbinian Staudacher, Tobias Guggemos, Sophia Grundner-Culemann, and Wolfgang Gehrke, "Reducing 2-QuBit Gate Count for ZX-Calculus based Quantum Circuit Optimization", arXiv:2311.08881, (2023).

[198] Nathan Claudet and Simon Perdrix, "Covering a Graph with Minimal Local Sets", arXiv:2402.10678, (2024).

[199] Mehdi Mhalla, Simon Perdrix, and Luc Sanselme, "Shadow Pauli Flow: Characterising Determinism in MBQCs involving Pauli Measurements", arXiv:2207.09368, (2022).

[200] Titouan Carette, Dominic Horsman, and Simon Perdrix, "SZX-calculus: Scalable Graphical Quantum Reasoning", arXiv:1905.00041, (2019).

[201] John van de Wetering and Sal Wolffs, "Completeness of the Phase-free ZH-calculus", arXiv:1904.07545, (2019).

[202] Alexis Toumi, "Category Theory for Quantum Natural Language Processing", arXiv:2212.06615, (2022).

[203] Tobias Stollenwerk and Stuart Hadfield, "Measurement-Based Quantum Approximate Optimization", arXiv:2403.11514, (2024).

[204] Raban Iten, Romain Moyard, Tony Metger, David Sutter, and Stefan Woerner, "Exact and practical pattern matching for quantum circuit optimization", arXiv:1909.05270, (2019).

[205] Narayanan Rengaswamy, "Classical Coding Approaches to Quantum Applications", arXiv:2004.06834, (2020).

[206] Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova, "Addition and Differentiation of ZX-diagrams", arXiv:2202.11386, (2022).

[207] Shuxiang Cao, "Multi-agent blind quantum computation without universal cluster states", New Journal of Physics 25 10, 103028 (2023).

[208] Waldemir Cambiucci, Regina Melo Silveira, and Wilson Vicente Ruggiero, "Hypergraphic partitioning of quantum circuits for distributed quantum computing", arXiv:2301.05759, (2023).

[209] Mingsheng Ying, Li Zhou, and Gilles Barthe, "Laws of Quantum Programming", arXiv:2412.19463, (2024).

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