The ZX calculus is a language for surface code lattice surgery

Niel de Beaudrap; Dominic Horsman

doi:10.22331/q-2020-01-09-218

The ZX calculus is a language for surface code lattice surgery

¹Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD
²Department of Physics, Durham University, South Road, Durham, DH1 1LE Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD

Published:	2020-01-09, volume 4, page 218
Eprint:	arXiv:1704.08670v4
Doi:	https://doi.org/10.22331/q-2020-01-09-218
Citation:	Quantum 4, 218 (2020).

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Updated after initial publication: This publication was updated to version v4 after the initial publication. The authors left the following comment on the arXiv:

20 pages, many figures. Minor revisions. Accepted to Quantum Journal

Abstract

A leading choice of error correction for scalable quantum computing is the surface code with lattice surgery. The basic lattice surgery operations, the merging and splitting of logical qubits, act non-unitarily on the logical states and are not easily captured by standard circuit notation. This raises the question of how best to design, verify, and optimise protocols that use lattice surgery, in particular in architectures with complex resource management issues. In this paper we demonstrate that the operations of the ZX calculus --- a form of quantum diagrammatic reasoning based on bialgebras --- match exactly the operations of lattice surgery. Red and green ``spider'' nodes match rough and smooth merges and splits, and follow the axioms of a dagger special associative Frobenius algebra. Some lattice surgery operations require non-trivial correction operations, which are captured natively in the use of the ZX calculus in the form of ensembles of diagrams. We give a first taste of the power of the calculus as a language for lattice surgery by considering two operations (T gates and producing a CNOT) and show how ZX diagram re-write rules give lattice surgery procedures for these operations that are novel, efficient, and highly configurable.

► BibTeX data

@article{deBeaudrap2020zxcalculusis,
  doi = {10.22331/q-2020-01-09-218},
  url = {https://doi.org/10.22331/q-2020-01-09-218},
  title = {The {ZX} calculus is a language for surface code lattice surgery},
  author = {de Beaudrap, Niel and Horsman, Dominic},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {218},
  month = jan,
  year = {2020}
}

► References

[1] M. Backens. The ZX-calculus is complete for stabilizer quantum mechanics. New Journal of Physics, 16 (9): 093021, 2014. 10.1088/1367-2630/16/9/093021.
https://doi.org/10.1088/1367-2630/16/9/093021

[2] Miriam Backens. Making the stabilizer ZX-calculus complete for scalars. In Chris Heunen, Peter Selinger, and Jamie Vicary, editors, Proceedings of the 12th International Workshop on Quantum Physics and Logic (QPL 2015), volume 195 of Electronic Proceedings in Theoretical Computer Science, pages 17–32, 2015. 10.4204/EPTCS.195.2.
https://doi.org/10.4204/EPTCS.195.2

[3] H. Bombin. Topological order with a twist: Ising anyons from an Abelian model. Physical review letters, 105 (3): 030403, 2010. 10.1103/PhysRevLett.105.030403.
https://doi.org/10.1103/PhysRevLett.105.030403

[4] H. Bombin and M. Martin-Delgado. Quantum measurements and gates by code deformation. Journal of Physics A: Mathematical and Theoretical, 42 (9): 095302, 2009. 10.1088/1751-8113/42/9/095302.
https://doi.org/10.1088/1751-8113/42/9/095302

[5] S. Bravyi and A. Kitaev. Quantum codes on a lattice with boundary. Preprint, arXiv:quant-ph/9811052, 1998. Translation of Quantum Computers and Computing 2 (1), pp. 43-48. (2001).
arXiv:quant-ph/9811052

[6] S. Bravyi and A. Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A, 71 (2): 022316, 2005. 10.1103/PhysRevA.71.022316.
https://doi.org/10.1103/PhysRevA.71.022316

[7] E. Campbell and J. O'Gorman. An efficient magic state approach to small angle rotations. Quantum Science and Technology, 1 (1): 015007, 2016. 10.1088/2058-9565/1/1/015007.
https://doi.org/10.1088/2058-9565/1/1/015007

[8] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549 (7671): 172, 2017. 10.1038/nature23460. arXiv:1612.07330.
https://doi.org/10.1038/nature23460
arXiv:1612.07330

[9] T. Carette, D. Horsman, and S. Perdrix. SZX-calculus: Scalable graphical quantum reasoning. Preprint, arXiv:1905.00041, 2019.
arXiv:1905.00041

[10] Nicholas Chancellor, Aleks Kissinger, Joschka Roffe, Stefan Zohren, and Dominic Horsman. Graphical structures for design and verification of quantum error correction. arXiv:1611.08012, 2018.
arXiv:1611.08012

[11] B. Coecke and R. Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13 (4): 043016, 2011. 10.1088/1367-2630/13/4/043016.
https://doi.org/10.1088/1367-2630/13/4/043016

[12] B. Coecke and A. Kissinger. Picturing Quantum Processes: A first course in quantum theory and diagrammatic reasoning. Cambridge University Press, 2017. 10.1017/9781316219317.
https://doi.org/10.1017/9781316219317

[13] B Coecke, E. Paquette, and D. Pavlovic. Classical and quantum structuralism. Semantic Techniques in Quantum Computation, eds. Gay S., Mackie I., Cambridge University Press, 2008. 10.1017/CBO9781139193313.003. arXiv:0904.1997.
https://doi.org/10.1017/CBO9781139193313.003
arXiv:0904.1997

[14] Alexander Cowtan, Silas Dilkes, Ross Duncan, Alexandre Krajenbrink, Will Simmons, and Seyon Sivarajah. On the Qubit Routing Problem. In Wim van Dam and Laura Mancinska, editors, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019), volume 135 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1–5:32, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. ISBN 978-3-95977-112-2. 10.4230/LIPIcs.TQC.2019.5. URL http://drops.dagstuhl.de/opus/volltexte/2019/10397. arXiv:1902.08091.
https://doi.org/10.4230/LIPIcs.TQC.2019.5
arXiv:1902.08091
http://drops.dagstuhl.de/opus/volltexte/2019/10397

[15] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill. Topological quantum memory. Journal of Mathematical Physics, 43 (9): 4452–4505, 2002. 10.1063/1.1499754. arXiv:quant-ph/0110143.
https://doi.org/10.1063/1.1499754
arXiv:quant-ph/0110143

[16] R. Duncan and S. Perdrix. Graph states and the necessity of Euler decomposition. In Conference on Computability in Europe, pages 167–177. Springer, 2009. 10.1007/978-3-642-03073-4.
https://doi.org/10.1007/978-3-642-03073-4

[17] Ross Duncan and Maxime Lucas. Verifying the steane code with Quantomatic. Proceedings QPL 2013, pages 33–49, 2013. 10.4204/EPTCS.171.4. arXiv:1306.4532.
https://doi.org/10.4204/EPTCS.171.4
arXiv:1306.4532

[18] Ross Duncan and Simon Perdrix. Rewriting measurement-based quantum computations with generalised flow. In Samson Abramsky, Cyril Gavoille, Claude Kirchner, Friedhelm Meyer auf der Heide, and Paul G. Spirakis, editors, Automata, Languages and Programming, pages 285–296, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. ISBN 978-3-642-14162-1.

[19] Ross Duncan, Aleks Kissinger, Simon Perdrix, and John van de Wetering. Graph-theoretic simplification of quantum circuits with the ZX-calculus. Preprint, arXiv:1902.03178, 2019.
arXiv:1902.03178

[20] B. Eastin and E. Knill. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett., 102: 110502, Mar 2009. 10.1103/PhysRevLett.102.110502. arXiv:0811.4262.
https://doi.org/10.1103/PhysRevLett.102.110502
arXiv:0811.4262

[21] Andrew Fagan and Ross Duncan. Optimising Clifford circuits with Quantomatic. In Proceedings of the 15th International Conference on Quantum Physics and Logic (QPL), volume 287 of Electronic Proceedings in Theoretical Computer Science, pages 85–105. Open Publishing Association, 2019. 10.4204/EPTCS.287.5.
https://doi.org/10.4204/EPTCS.287.5

[22] A. Fowler, A. Stephens, and P. Groszkowski. High-threshold universal quantum computation on the surface code. Phys. Rev. A, 80: 052312, 2009. 10.1103/PhysRevA.80.052312.
https://doi.org/10.1103/PhysRevA.80.052312

[23] Austin G Fowler and Craig Gidney. Low overhead quantum computation using lattice surgery. Preprint, arXiv:1808.06709, 2018.
arXiv:1808.06709

[24] M. Freedman and D. Meyer. Projective plane and planar quantum codes. Foundations of Computational Mathematics, 1 (3): 325–332, 2001. 10.1007/s102080010013.
https://doi.org/10.1007/s102080010013

[25] Craig Gidney and Austin G Fowler. Efficient magic state factories with a catalyzed $\mathrm{\lvert CCZ\rangle}$ to $\mathrm{2\lvert T\rangle}$ transformation. Quantum, 3, 2019. 10.22331/q-2019-04-30-135. arXiv:1812.01238.
https://doi.org/10.22331/q-2019-04-30-135
arXiv:1812.01238

[26] Google. https://ai.googleblog.com/2018/03/a-preview-of-bristlecone-googles-new.html. Accessed 10/04/2019.
https://ai.googleblog.com/2018/03/a-preview-of-bristlecone-googles-new.html

[27] Daniel Gottesman. Class of quantum error-correcting codes saturating the quantum Hamming bound. Physical Review A, 54 (3): 1862, 1996. 10.1103/PhysRevA.54.1862. arXiv:quant-ph/9604038.
https://doi.org/10.1103/PhysRevA.54.1862
arXiv:quant-ph/9604038

[28] Amar Hadzihasanovic, Kang Feng Ng, and Quanlong Wang. Two complete axiomatisations of pure-state qubit quantum computing. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '18, pages 502–511, New York, NY, USA, 2018. ACM. ISBN 978-1-4503-5583-4. 10.1145/3209108.3209128.
https://doi.org/10.1145/3209108.3209128

[29] D. Herr, F. Nori, and S. Devitt. Lattice surgery translation for quantum computation. New Journal of Physics, (19): 013034, 2017. 10.1088/1367-2630/aa5709.
https://doi.org/10.1088/1367-2630/aa5709

[30] C. Horsman, A. G Fowler, S. Devitt, and R. Van Meter. Surface code quantum computing by lattice surgery. New Journal of Physics, 14 (12): 123011, 2012. 10.1088/1367-2630/14/12/123011. arXiv:1111.4022.
https://doi.org/10.1088/1367-2630/14/12/123011
arXiv:1111.4022

[31] IBM. https://www.research.ibm.com/ibm-q/. Accessed 10/04/2019.
https://www.research.ibm.com/ibm-q/

[32] C. Jones, D. Kim, M. Rakher, P. Kwiat, and T. Ladd. Design and analysis of communication protocols for quantum repeater networks. New Journal of Physics, 18 (8): 083015, 2016. 10.1088/1367-2630/18/8/083015.
https://doi.org/10.1088/1367-2630/18/8/083015

[33] A. Kissinger and V. Zamdzhiev. Quantomatic: A proof assistant for diagrammatic reasoning. In International Conference on Automated Deduction, pages 326–336. Springer, 2015. 10.1007/978-3-319-21401-6_22.
https://doi.org/10.1007/978-3-319-21401-6_22

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

[35] Aleks Kissinger and John van de Wetering. Reducing T-count with the ZX-calculus. Preprint, arXiv:1903.10477, 2019a.
arXiv:1903.10477

[36] Aleks Kissinger and John van de Wetering. PyZX: Large scale automated diagrammatic reasoning. Preprint, arXiv:1904.04735, 2019b.
arXiv:1904.04735

[37] E. Knill. Quantum computing with realistically noisy devices. Nature, 434 (7029): 39–44, 2005. 10.1038/nature03350. arXiv:quant-ph/0410199.
https://doi.org/10.1038/nature03350
arXiv:quant-ph/0410199

[38] Andrew J Landahl and Ciáran Ryan-Anderson. Quantum computing by color-code lattice surgery. Preprint, arXiv:1407.5103, 2014. SAND2014-15911J.
arXiv:1407.5103

[39] Daniel Litinski. A game of surface codes: Large-scale quantum computing with lattice surgery. Quantum, 3: 128, 2019. 10.22331/q-2019-03-05-128.
https://doi.org/10.22331/q-2019-03-05-128

[40] N. Nickerson, Y. Li, and S. Benjamin. Topological quantum computing with a very noisy network and local error rates approaching one percent. Nature communications, 4: 1756, 2013. 10.1038/ncomms2773.
https://doi.org/10.1038/ncomms2773

[41] N. Nickerson, J. Fitzsimons, and S. Benjamin. Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links. Physical Review X, 4 (4): 041041, 2014. 10.1103/PhysRevX.4.041041.
https://doi.org/10.1103/PhysRevX.4.041041

[42] Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information. Cambridge University Press, Cambridge UK, 2000. 10.1017/CBO9780511976667.
https://doi.org/10.1017/CBO9780511976667

[43] R. Raussendorf and J. Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Physical review letters, 98 (19): 190504, 2007. 10.1103/PhysRevLett.98.190504.
https://doi.org/10.1103/PhysRevLett.98.190504

[44] B. Terhal. Quantum error correction for quantum memories. Reviews of Modern Physics, 87 (2): 307, 2015. 10.1103/RevModPhys.87.307.
https://doi.org/10.1103/RevModPhys.87.307

[45] R. Van Meter. Quantum Networking. John Wiley & Sons, 2014. 10.1002/9781118648919.
https://doi.org/10.1002/9781118648919

[46] R. Van Meter and C. Horsman. A blueprint for building a quantum computer. Communications of the ACM, 56 (10): 84–93, 2013. 10.1145/2494568.
https://doi.org/10.1145/2494568

[47] The ZX-calculus. http://zxcalculus.com/. Accessed 08/07/2019.
http://zxcalculus.com/

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[1] 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).

[2] Selma Dündar-Coecke, Lia Yeh, Emmanuel Pothos, Muhammad Hamza Waseem, and Bob Coecke, "Beyond symbolic algebra with quantum picturalism", Frontiers in Cognition 5, 1790789 (2026).

[3] Niel de Beaudrap, Ross Duncan, Dominic Horsman, and Simon Perdrix, "Pauli Fusion: a Computational Model to Realise Quantum Transformations from ZX Terms", Electronic Proceedings in Theoretical Computer Science 318, 85 (2020).

[4] 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.

[5] Nicholas Chancellor, Aleks Kissinger, Stefan Zohren, Joschka Roffe, and Dominic Horsman, "Graphical structures for design and verification of quantum error correction", Quantum Science and Technology 8 4, 045028 (2023).

[6] Ying Li, "Low-density parity-check representation of fault-tolerant quantum circuits", Physical Review Research 7 1, 013115 (2025).

[7] Mithün Paul, Tiju Cherian John, and Nithin Raveendran, Communications in Computer and Information Science 2659, 93 (2026) ISBN:978-3-032-13882-8.

[8] 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).

[9] Tyler Leblond, Ryan S. Bennink, Justin G. Lietz, and Christopher M. Seck, Proceedings of the SC '23 Workshops of the International Conference on High Performance Computing, Network, Storage, and Analysis 1426 (2023) ISBN:9798400707858.

[10] 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).

[11] Michael Hanks, Marta P. Estarellas, William J. Munro, and Kae Nemoto, "Effective Compression of Quantum Braided Circuits Aided by ZX-Calculus", Physical Review X 10 4, 041030 (2020).

[12] Pranay Gorantla, Shu-Heng Shao, and Nathanan Tantivasadakarn, "Tensor Networks for Noninvertible Symmetries in 3+1D and Beyond", Physical Review X 15 4, 041006 (2025).

[13] Joseph Collins and Ross Duncan, "Hopf-Frobenius Algebras and a Simpler Drinfeld Double", Electronic Proceedings in Theoretical Computer Science 318, 150 (2020).

[14] Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, and Sal Wolffs, "Completeness of the ZH-calculus", Compositionality 5, 5 (2023).

[15] Alejandro Villoria, Henning Basold, and Alfons Laarman, Lecture Notes in Computer Science 14574, 121 (2024) ISBN:978-3-031-57227-2.

[16] Adrian Lehmann, Ben Caldwell, Bhakti Shah, William Spencer, and Robert Rand, " Vy ZX: Formal Verification of a Graphical Quantum Language ", ACM Transactions on Programming Languages and Systems 48 3, 1 (2026).

[17] 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).

[18] Rihan Hai, Shih-Han Hung, Tim Coopmans, Tim Littau, and Floris Geerts, "Quantum Data Management in the NISQ Era", Proceedings of the VLDB Endowment 18 6, 1720 (2025).

[19] Avimita Chatterjee, Subrata Das, and Swaroop Ghosh, "Lattice Surgery for Dummies", Sensors 25 6, 1854 (2025).

[20] Dong-Xiao Quan, Xiao-Jie Lü, and Wen-Fei Zhang, "Structure design and logical CNOT implementation of multi-logical-qubits surface code", Acta Physica Sinica 73 4, 040304 (2024).

[21] Tuomas Laakkonen, Konstantinos Meichanetzidis, and John van de Wetering, "Picturing Counting Reductions with the ZH-Calculus", Electronic Proceedings in Theoretical Computer Science 384, 89 (2023).

[22] Quanlong Wang, Richie Yeung, and Mark Koch, "Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning", Quantum 8, 1491 (2024).

[23] Giovanni de Felice, Boldizsár Poór, Cole Comfort, Lia Yeh, Mateusz Kupper, William Cashman, and Bob Coecke, "A dataflow programming framework for linear optical distributed quantum computing", Quantum 10, 1972 (2026).

[24] Boldizsár Poór, Robert I. Booth, Titouan Carette, John van de Wetering, and Lia Yeh, "The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification", Electronic Proceedings in Theoretical Computer Science 384, 220 (2023).

[25] Hector Bombin, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, and Sam Roberts, "Unifying flavors of fault tolerance with the ZX calculus", Quantum 8, 1379 (2024).

[26] Felix Thomsen, Markus S. Kesselring, Stephen D. Bartlett, and Benjamin J. Brown, "Low-overhead quantum computing with the color code", Physical Review Research 6 4, 043125 (2024).

[27] Vladlen Galetsky, Nilesh Vyas, Alberto Comin, and Janis Nötzel, 2025 International Conference on Quantum Communications, Networking, and Computing (QCNC) 677 (2025) ISBN:979-8-3315-3159-1.

[28] Fatimah Rita Ahmadi and Aleks Kissinger, "The ZX-calculus as a language for topological quantum computation", Journal of Physics A: Mathematical and Theoretical 56 41, 415301 (2023).

[29] 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.

[30] 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).

[31] 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).

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

[33] Yuxiang Liu, Zaichen Zhang, Yi Hu, Fanxu Meng, Tian Luan, Xianchao Zhang, and Xutao Yu, "Practical circuit optimization algorithm for quantum simulation based on template matching", Quantum Information Processing 23 2, 45 (2024).

[34] Craig Gidney, Michael Newman, Peter Brooks, and Cody Jones, "Yoked surface codes", Nature Communications 16 1, 4498 (2025).

[35] Mateusz Kupper, Dominic Horsman, Chris Heunen, and Niel de Beaudrap, "String Diagrams for Defect-Based Surface Code Computing", Electronic Proceedings in Theoretical Computer Science 426, 159 (2025).

[36] Boldizsár Poór, Quanlong Wang, Razin A. Shaikh, Lia Yeh, Richie Yeung, and Bob Coecke, 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 1 (2023) ISBN:979-8-3503-3587-3.

[37] Daniel Bochen Tan, Murphy Yuezhen Niu, and Craig Gidney, 2024 ACM/IEEE 51st Annual International Symposium on Computer Architecture (ISCA) 325 (2024) ISBN:979-8-3503-2658-1.

[38] Hector Miller-Bakewell, "Finite Verification of Infinite Families of Diagram Equations", Electronic Proceedings in Theoretical Computer Science 318, 27 (2020).

[39] Moritz Lange, Pontus Havström, Basudha Srivastava, Isak Bengtsson, Valdemar Bergentall, Karl Hammar, Olivia Heuts, Evert van Nieuwenburg, and Mats Granath, "Data-driven decoding of quantum error correcting codes using graph neural networks", Physical Review Research 7 2, 023181 (2025).

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

[41] Craig Gidney, "A Pair Measurement Surface Code on Pentagons", Quantum 7, 1156 (2023).

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

[43] Andrey Boris Khesin, Jonathan Z. Lu, and Peter W. Shor, "Universal Graph Representation of Stabilizer Codes", PRX Quantum 6 4, 040325 (2025).

[44] John H. Selby, Carlo Maria Scandolo, and Bob Coecke, "Reconstructing quantum theory from diagrammatic postulates", Quantum 5, 445 (2021).

[45] Augustin Borgna and Rafael Romero, "Encoding High-level Quantum Programs as SZX-diagrams", Electronic Proceedings in Theoretical Computer Science 394, 141 (2023).

[46] Jiahan Chen, Chenghong Zhu, Ge Bai, and Xin Wang, 2026 ACM/IEEE 53rd Annual International Symposium on Computer Architecture (ISCA) 842 (2026) ISBN:979-8-3315-5065-3.

[47] Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, Fernando Pastawski, and Sam Roberts, "Logical Blocks for Fault-Tolerant Topological Quantum Computation", PRX Quantum 4 2, 020303 (2023).

[48] Vladlen Galetsky, Nilesh Vyas, Alberto Comin, and Janis Nötzel, "Feasibility of logical Bell state generation in memory assisted quantum networks", Physical Review Research 7 3, 033090 (2025).

[49] Filipa C R Peres, Rafael Wagner, and Ernesto F Galvão, "Non-stabilizerness and entanglement from cat-state injection", New Journal of Physics 26 1, 013051 (2024).

[50] Chien-Tung Cherie Kuo, Cheng-En Tsai, and Chung-Yang Ric Huang, 2026 Design, Automation & Test in Europe Conference (DATE) 1 (2026) ISBN:978-3-9826741-1-7.

[51] Alexander Cowtan and Simon Burton, "CSS code surgery as a universal construction", Quantum 8, 1344 (2024).

[52] Chen Zhao and Xiao-Shan Gao, "Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus", Quantum 5, 466 (2021).

[53] Craig Gidney, "How to factor 2048 bit RSA integers with less than a million noisy qubits", arXiv:2505.15917, (2025).

[54] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[55] Craig Gidney and Austin G. Fowler, "Efficient magic state factories with a catalyzed|CCZ⟩to2|T⟩transformation", Quantum 3, 135 (2019).

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

[57] Craig Gidney and Austin G. Fowler, "Flexible layout of surface code computations using AutoCCZ states", arXiv:1905.08916, (2019).

[58] Renaud Vilmart, "A Near-Optimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics", arXiv:1812.09114, (2018).

[59] Benjamin Rodatz, Boldizsár Poór, and Aleks Kissinger, "Floquetifying stabiliser codes with distance-preserving rewrites", arXiv:2410.17240, (2024).

[60] Christophe Vuillot, Lingling Lao, Ben Criger, Carmen García Almudéver, Koen Bertels, and Barbara M. Terhal, "Code deformation and lattice surgery are gauge fixing", New Journal of Physics 21 3, 033028 (2019).

[61] Daniel Bochen Tan, Murphy Yuezhen Niu, and Craig Gidney, "A SAT Scalpel for Lattice Surgery: Representation and Synthesis of Subroutines for Surface-Code Fault-Tolerant Quantum Computing", arXiv:2404.18369, (2024).

[62] 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).

[63] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "Completeness of the ZX-Calculus", arXiv:1903.06035, (2019).

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

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

[66] Alexander Cowtan, Zhiyang He, Dominic J. Williamson, and Theodore J. Yoder, "Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery", arXiv:2510.14895, (2025).

[67] Sam McArdle, Alexander M. Dalzell, Aleksander Kubica, and Fernando G. S. L. Brandão, "The Fast for the Curious: How to accelerate fault-tolerant quantum applications", arXiv:2510.26078, (2025).

[68] Bob Coecke, Stefano Gogioso, and John H. Selby, "The time-reverse of any causal theory is eternal noise", arXiv:1711.05511, (2017).

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

[70] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics", arXiv:1705.11151, (2017).

[71] Selma Dündar-Coecke, Caterina Puca, Lia Yeh, Muhammad Hamza Waseem, Emmanuel M. Pothos, Thomas Cervoni, Sieglinde M.-L. Pfaendler, Vincent Wang-Maścianica, Peter Sigrist, Ferdi Tomassini, Vincent Anandraj, Ilyas Khan, Stefano Gogioso, Aleks Kissinger, and Bob Coecke, "Making the quantum world accessible to young learners through Quantum Picturalism: An experimental study", arXiv:2504.01013, (2025).

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

[73] Renaud Vilmart, "A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond", arXiv:1804.03084, (2018).

[74] Quanlong Wang, Boldizsár Poór, and Razin A. Shaikh, "Completeness of qufinite ZXW calculus, a graphical language for finite-dimensional quantum theory", arXiv:2309.13014, (2023).

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

[76] Boldizsár Poór, Quanlong Wang, Razin A. Shaikh, Lia Yeh, Richie Yeung, and Bob Coecke, "Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus", arXiv:2302.12135, (2023).

[77] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "Diagrammatic Reasoning beyond Clifford+T Quantum Mechanics", arXiv:1801.10142, (2018).

[78] William J. Huggins, Tanuj Khattar, Amanda Xu, Matthew Harrigan, Christopher Kang, Guang Hao Low, Austin Fowler, Nicholas C. Rubin, and Ryan Babbush, "The FLuid Allocation of Surface code Qubits (FLASQ) cost model for early fault-tolerant quantum algorithms", arXiv:2511.08508, (2025).

[79] Richard D. P. East, Pierre Martin-Dussaud, and John Van de Wetering, "Spin-networks in the ZX-calculus", arXiv:2111.03114, (2021).

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

[81] Bob Coecke, "Basic ZX-calculus for students and professionals", arXiv:2303.03163, (2023).

[82] 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).

[83] Quanlong Wang, Richard D. P. East, Razin A. Shaikh, Lia Yeh, Boldizsár Poór, and Bob Coecke, "Beyond Penrose tensor diagrams with the ZX calculus: Applications to quantum computing, quantum machine learning, condensed matter physics, and quantum gravity", arXiv:2511.06012, (2025).

[84] Marc de Visme and Renaud Vilmart, "Minimality in Finite-Dimensional ZW-Calculi", arXiv:2401.16225, (2024).

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

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

[87] Haytham McDowall-Rose, Razin A. Shaikh, and Lia Yeh, "From Fermions to Qubits: A ZX-Calculus Perspective", arXiv:2505.06212, (2025).

[88] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

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

[90] Bob Coecke and Quanlong Wang, "ZX-Rules for 2-qubit Clifford+T Quantum Circuits", arXiv:1804.05356, (2018).

[91] Adrian Lehmann, Ben Caldwell, Bhakti Shah, William Spencer, and Robert Rand, "VyZX: Formal Verification of a Graphical Quantum Language", arXiv:2311.11571, (2023).

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

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

[94] Robert I. Booth, Titouan Carette, and Cole Comfort, "Complete equational theories for classical and quantum Gaussian relations", arXiv:2403.10479, (2024).

[95] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness", arXiv:1805.05296, (2018).

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

[97] Stefano Gogioso and Subhayan Roy Moulik, "Purification and time-reversal deny entanglement in LOCC-distinguishable orthonormal bases", arXiv:1902.00316, (2019).

[98] Benoît Valiron, "On Quantum Programming Languages", arXiv:2410.13337, (2024).

[99] Quanlong Wang, "Completeness of the ZX-calculus", arXiv:2209.14894, (2022).

[100] Tuomas Laakkonen, Konstantinos Meichanetzidis, and John van de Wetering, "Picturing Counting Reductions with the ZH-Calculus", arXiv:2304.02524, (2023).

[101] Augustin Borgna and Rafael Romero, "Encoding High-level Quantum Programs as SZX-diagrams", arXiv:2206.09376, (2022).

[102] Boldizsár Poór, Robert I. Booth, Titouan Carette, John van de Wetering, and Lia Yeh, "The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification", arXiv:2306.05204, (2023).

[103] Renaud Vilmart, "Quantum Multiple-Valued Decision Diagrams in Graphical Calculi", arXiv:2107.01186, (2021).

[104] Alejandro Villoria, Henning Basold, and Alfons Laarman, "Enriching Diagrams with Algebraic Operations", arXiv:2310.11288, (2023).

[105] Laura S. Herzog, Gilad Kishony, Robert Wille, and Austin Fowler, "Towards Lattice Surgery Compilation for the Color Code Using Pipe Diagrams", arXiv:2607.05501, (2026).

[106] Valter Uotila, Cong Yu, and Bo Zhao, "ZX-DB: A Graph Database for Quantum Circuit Simplification and Rewriting via the ZX-Calculus", arXiv:2511.13033, (2025).

[107] Giovanni de Felice, Boldizsár Poór, Cole Comfort, Lia Yeh, Mateusz Kupper, William Cashman, and Bob Coecke, "A dataflow programming framework for linear optical distributed quantum computing", arXiv:2601.08389, (2026).

[108] Steven Rayan, "A diagrammatic field theory of quantum error correction", arXiv:2607.08911, (2026).

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