Unifying flavors of fault tolerance with the ZX calculus

Hector Bombin; Daniel Litinski; Naomi Nickerson; Fernando Pastawski; Sam Roberts

doi:10.22331/q-2024-06-18-1379

Unifying flavors of fault tolerance with the ZX calculus

Hector Bombin, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, and Sam Roberts

PsiQuantum, Palo Alto

Published:	2024-06-18, volume 8, page 1379
Eprint:	arXiv:2303.08829v3
Doi:	https://doi.org/10.22331/q-2024-06-18-1379
Citation:	Quantum 8, 1379 (2024).

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Abstract

There are several models of quantum computation which exhibit shared fundamental fault-tolerance properties. This article makes commonalities explicit by presenting these different models in a unifying framework based on the ZX calculus. We focus on models of topological fault tolerance – specifically surface codes – including circuit-based, measurement-based and fusion-based quantum computation, as well as the recently introduced model of Floquet codes. We find that all of these models can be viewed as different flavors of the same underlying stabilizer fault-tolerance structure, and sustain this through a set of local equivalence transformations which allow mapping between flavors. We anticipate that this unifying perspective will pave the way to transferring progress among the different views of stabilizer fault-tolerance and help researchers familiar with one model easily understand others.

Featured image: The shared backbone of surface-code style stabilizer fault-tolerance is represented using the diagrammatic notation of the ZX calculus (graph with green and red nodes). Features specific to each compuational paradigm are highlighted such as physical qubit world-lines in the circuit model (red) or floquet model (blue) and resource states for fusion based model (purple).

Popular summary

There are different models of quantum computation such as circuit-based, measurement-based, fusion-based. While each model is based on different elementary primitives , they are know to have the same expressive power. For each of these models, fault-tolerance allows implementing reliable quantum computation when these primitives are unreliable. This article shows how a fault-tolerant realization in each of these models can be expressed in the diagrammatic language of the ZX calculus, a graphical notation based on very few elements. Doing so, makes the common fault-tolerance structure self-evident. This unifying perspective aims to facilitate knowledge transfer and understanding across different paradigms, potentially accelerating progress in quantum computing research.

► BibTeX data

@article{Bombin2024unifyingflavorsof,
  doi = {10.22331/q-2024-06-18-1379},
  url = {https://doi.org/10.22331/q-2024-06-18-1379},
  title = {Unifying flavors of fault tolerance with the {ZX} calculus},
  author = {Bombin, Hector and Litinski, Daniel and Nickerson, Naomi and Pastawski, Fernando and Roberts, Sam},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1379},
  month = jun,
  year = {2024}
}

► References

[1] Alexei Yu Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of Physics 303, 2–30 (2003).
https://doi.org/10.1016/S0003-4916(02)00018-0

[2] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. Journal of Mathematical Physics 43, 4452–4505 (2002).
https://doi.org/10.1063/1.1499754

[3] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. ``Measurement-based quantum computation on cluster states''. Physical Review A 68, 022312 (2003).
https://doi.org/10.1103/PhysRevA.68.022312

[4] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van Den Nest. ``Measurement-based quantum computation''. Nature Physics 2009 5:1 5, 19–26 (2009).
https://doi.org/10.1038/nphys1157

[5] R. Raussendorf, J. Harrington, and K. Goyal. ``A fault-tolerant one-way quantum computer''. Annals of Physics 321, 2242–2270 (2006).
https://doi.org/10.1016/j.aop.2006.01.012

[6] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, et al. ``Fusion-based quantum computation''. Nature Communications 14, 912 (2023).
https://doi.org/10.1038/s41467-023-36493-1

[7] Matthew B. Hastings and Jeongwan Haah. ``Dynamically generated logical qubits''. Quantum 5, 564 (2021).
https://doi.org/10.22331/q-2021-10-19-564

[8] Adam Paetznick, Christina Knapp, Nicolas Delfosse, Bela Bauer, Jeongwan Haah, Matthew B. Hastings, and Marcus P. da Silva. ``Performance of planar Floquet codes with Majorana-based qubits''. PRX Quantum 4, 010310 (2023).
https://doi.org/10.1103/PRXQuantum.4.010310

[9] Jeongwan Haah and Matthew B. Hastings. ``Boundaries for the honeycomb code''. Quantum 6, 693 (2022).
https://doi.org/10.22331/q-2022-04-21-693

[10] Craig Gidney, Michael Newman, and Matt McEwen. ``Benchmarking the planar honeycomb code''. Quantum 6, 813 (2022).
https://doi.org/10.22331/q-2022-09-21-813

[11] Craig Gidney. ``A pair measurement surface code on pentagons''. Quantum 7, 1156 (2023).
https://doi.org/10.22331/q-2023-10-25-1156

[12] Markus S. Kesselring, Julio C. Magdalena de la Fuente, Felix Thomsen, Jens Eisert, Stephen D. Bartlett, and Benjamin J. Brown. ``Anyon condensation and the color code''. PRX Quantum 5, 010342 (2024).
https://doi.org/10.1103/PRXQuantum.5.010342

[13] Ramón Aguado and Leo P. Kouwenhoven. ``Majorana qubits for topological quantum computing''. Physics Today 73, 44 (2020).
https://doi.org/10.1063/PT.3.4499

[14] 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, 020303 (2023).
https://doi.org/10.1103/PRXQuantum.4.020303

[15] Daniel Gottesman. ``Opportunities and challenges in fault-tolerant quantum computation'' arXiv:2210.15844 (2022).
https://doi.org/10.48550/arXiv.2210.15844
arXiv:2210.15844

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

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

[18] Bob Coecke and Aleks Kissinger. ``Picturing quantum processes: A first course in quantum theory and diagrammatic reasoning''. Cambridge University Press. (2017).
https://doi.org/10.1017/9781316219317

[19] Niel de Beaudrap and Dominic Horsman. ``The ZX calculus is a language for surface code lattice surgery''. Quantum 4, 218 (2020).
https://doi.org/10.22331/q-2020-01-09-218

[20] Amar Hadzihasanovic, Kang Feng Ng, and Quanlong Wang. ``Two complete axiomatisations of pure-state qubit quantum computing''. Proceedings - Symposium on Logic in Computer SciencePages 502–511 (2018).
https://doi.org/10.1145/3209108.3209128

[21] Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. ``A complete axiomatisation of the ZX-calculus for Clifford+T quantum mechanics''. Proceedings - Symposium on Logic in Computer SciencePages 559–568 (2018).
https://doi.org/10.1145/3209108.3209131

[22] John van de Wetering. ``ZX-calculus for the working quantum computer scientist'' arXiv:2012.13966 (2020).
https://doi.org/10.48550/arXiv.2012.13966
arXiv:2012.13966

[23] 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 SystemsPages 1–1 (2022).
https://doi.org/10.1109/jetcas.2022.3202204

[24] Matt McEwen, Dave Bacon, and Craig Gidney. ``Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics''. Quantum 7, 1172 (2023).
https://doi.org/10.22331/q-2023-11-07-1172

[25] Margarita Davydova, Nathanan Tantivasadakarn, and Shankar Balasubramanian. ``Floquet codes without parent subsystem codes''. PRX Quantum 4, 020341 (2023).
https://doi.org/10.1103/PRXQuantum.4.020341

[26] Robert Raussendorf, Sergey Bravyi, and Jim Harrington. ``Long-range quantum entanglement in noisy cluster states''. Phys. Rev. A 71, 062313 (2005).
https://doi.org/10.1103/PhysRevA.71.062313

[27] Naomi Nickerson and Hector Bombin. ``Measurement based fault tolerance beyond foliation'' (2018).

[28] Hector Bombin, Isaac H. Kim, Daniel Litinski, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Sam Roberts, and Terry Rudolph. ``Interleaving: Modular architectures for fault-tolerant photonic quantum computing'' (2021).

[29] Michael Newman, Leonardo Andreta de Castro, and Kenneth R. Brown. ``Generating Fault-Tolerant Cluster States from Crystal Structures''. Quantum 4, 295 (2020).
https://doi.org/10.22331/q-2020-07-13-295

[30] H. Bombin and M. A. Martin-Delgado. ``Topological quantum distillation''. Phys. Rev. Lett. 97, 180501 (2006).
https://doi.org/10.1103/PhysRevLett.97.180501

[31] Hector Bombin. ``2D quantum computation with 3D topological codes'' arXiv:1810.09571 (2018).
https://doi.org/10.48550/arXiv.1810.09571
arXiv:1810.09571

[32] Nikolas P. Breuckmann and Jens Niklas Eberhardt. ``Quantum low-density parity-check codes''. PRX Quantum 2, 040101 (2021).
https://doi.org/10.1103/PRXQuantum.2.040101

[33] Andrey Boris Khesin, Jonathan Z Lu, and Peter W Shor. ``Graphical quantum Clifford-encoder compilers from the ZX calculus'' arXiv:2301.02356 (2023).
https://doi.org/10.48550/arXiv.2301.02356
arXiv:2301.02356

[34] Héctor Bombín, Chris Dawson, Ye-Hua Liu, Naomi Nickerson, Fernando Pastawski, and Sam Roberts. ``Modular decoding: parallelizable real-time decoding for quantum computers'' arXiv:2303.04846 (2023).
https://doi.org/10.48550/arXiv.2303.04846
arXiv:2303.04846

[35] Daniel Litinski and Naomi Nickerson. ``Active volume: An architecture for efficient fault-tolerant quantum computers with limited non-local connections'' arXiv:2211.15465 (2022).
https://doi.org/10.48550/arXiv.2211.15465
arXiv:2211.15465

Cited by

[1] Gilad Kishony and Erez Berg, "Increasing the distance of topological codes with time vortex defects", Quantum 10, 2006 (2026).

[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] Guo-Yi Zhu, Nathanan Tantivasadakarn, and Simon Trebst, "Structured volume-law entanglement in an interacting, monitored Majorana spin liquid", Physical Review Research 6 4, L042063 (2024).

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

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

[6] Kriti Sharma and Adarsh Kumar, 2025 2nd International Conference On Multidisciplinary Research and Innovations in Engineering (MRIE) 128 (2025) ISBN:979-8-3315-8673-7.

[7] Andreas Bauer, "x+y Floquet code: A simple example for topological quantum computation in the path-integral approach", Physical Review A 111 3, 032413 (2025).

[8] Cory T. Aitchison and Benjamin Béri, "Competing automorphisms and disordered Floquet codes", Physical Review B 111 23, 235112 (2025).

[9] Bhuvanesh Sundar, Bram Evert, Vasily Geyko, Andrew Patterson, Ilon Joseph, and Yuan Shi, "Simulating plasma wave propagation on a superconducting quantum chip", Physical Review Applied 25 2, 024077 (2026).

[10] Maike Ostmann, Joshua Nunn, and Alex E. Jones, "Nonlinear photonic architecture for fault-tolerant quantum computing", Physical Review A 114 1, 012618 (2026).

[11] Andreas Bauer, "Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases", Quantum 9, 1673 (2025).

[12] 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).

[13] Peter-Jan H.S. Derks, Alex Townsend-Teague, Ansgar G. Burchards, and Jens Eisert, "Designing fault-tolerant circuits using detector error models", Quantum 9, 1905 (2025).

[14] Margarita Davydova, Nathanan Tantivasadakarn, Shankar Balasubramanian, and David Aasen, "Quantum computation from dynamic automorphism codes", Quantum 8, 1448 (2024).

[15] Meng Sun, Bowen Yang, Zongyuan Wang, Nathanan Tantivasadakarn, and Yu-An Chen, "Clifford Quantum Cellular Automata from Topological Quantum Field Theories and Invertible Subalgebras", PRX Quantum 7 1, 010362 (2026).

[16] Yaodong Li, Nicholas O’Dea, and Vedika Khemani, "Perturbative Stability and Error-Correction Thresholds of Quantum Codes", PRX Quantum 6 1, 010327 (2025).

[17] Andreas Bauer and Julio C. Magdalena de la Fuente, "Planar Fault-Tolerant Circuits for Non-Clifford Gates on the 2D Color Code", PRX Quantum 7 2, 020367 (2026).

[18] 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).

[19] Rahul Sahay and Ruben Verresen, "Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements", PRX Quantum 6 1, 010329 (2025).

[20] 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).

[21] Lukas Voss, Sim Jian Xian, Tobias Haug, and Kishor Bharti, "Multivariate bicycle codes", Physical Review A 111 6, L060401 (2025).

[22] Amir-Reza Negari, Tyler D. Ellison, and Timothy H. Hsieh, "Spacetime Markov length: A diagnostic for fault tolerance via mixed-state phases", Physical Review B 114 4, 045130 (2026).

[23] Alec Eickbusch, Matt McEwen, Volodymyr Sivak, Alexandre Bourassa, Juan Atalaya, Jahan Claes, Dvir Kafri, Craig Gidney, Christopher W. Warren, Jonathan Gross, Alex Opremcak, Nicholas Zobrist, Kevin C. Miao, Gabrielle Roberts, Kevin J. Satzinger, Andreas Bengtsson, Matthew Neeley, William P. Livingston, Alex Greene, Rajeev Acharya, Laleh Aghababaie Beni, Georg Aigeldinger, Ross Alcaraz, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Ryan Babbush, Brian Ballard, Joseph C. Bardin, Alexander Bilmes, Jenna Bovaird, Dylan Bowers, Leon Brill, Michael Broughton, David A. Browne, Brett Buchea, Bob B. Buckley, Tim Burger, Brian Burkett, Nicholas Bushnell, Anthony Cabrera, Juan Campero, Hung-Shen Chang, Ben Chiaro, Liang-Ying Chih, Agnetta Y. Cleland, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Sayan Das, Alexander Del Toro Barba, Sean Demura, Laura De Lorenzo, Agustin Di Paolo, Paul Donohoe, Ilya K. Drozdov, Andrew Dunsworth, Aviv Moshe Elbag, Mahmoud Elzouka, Catherine Erickson, Vinicius S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, Suhas Ganjam, Gonzalo Garcia, Robert Gasca, Élie Genois, William Giang, Dar Gilboa, Raja Gosula, Alejandro Grajales Dau, Dietrich Graumann, Tan Ha, Steve Habegger, Michael C. Hamilton, Monica Hansen, Matthew P. Harrigan, Sean D. Harrington, Stephen Heslin, Paula Heu, Oscar Higgott, Reno Hiltermann, Jeremy Hilton, Hsin-Yuan Huang, Ashley Huff, William J. Huggins, Evan Jeffrey, Zhang Jiang, Xiaoxuan Jin, Cody Jones, Chaitali Joshi, Pavol Juhas, Andreas Kabel, Hui Kang, Amir H. Karamlou, Kostyantyn Kechedzhi, Trupti Khaire, Tanuj Khattar, Mostafa Khezri, Seon Kim, Bryce Kobrin, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, Vladislav D. Kurilovich, David Landhuis, Tiano Lange-Dei, Brandon W. Langley, Kim-Ming Lau, Justin Ledford, Kenny Lee, Brian J. Lester, Loïck Le Guevel, Wing Yan Li, Alexander T. Lill, Aditya Locharla, Erik Lucero, Daniel Lundahl, Aaron Lunt, Sid Madhuk, Ashley Maloney, Salvatore Mandrà, Leigh S. Martin, Orion Martin, Cameron Maxfield, Jarrod R. McClean, Seneca Meeks, Anthony Megrant, Reza Molavi, Sebastian Molina, Shirin Montazeri, Ramis Movassagh, Michael Newman, Anthony Nguyen, Murray Nguyen, Chia-Hung Ni, Logan Oas, Raymond Orosco, Kristoffer Ottosson, Alex Pizzuto, Rebecca Potter, Orion Pritchard, Chris Quintana, Ganesh Ramachandran, Matthew J. Reagor, David M. Rhodes, Eliott Rosenberg, Elizabeth Rossi, Kannan Sankaragomathi, Henry F. Schurkus, Michael J. Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Spencer Small, W. Clarke Smith, Sofia Springer, George Sterling, Jordan Suchard, Aaron Szasz, Alex Sztein, Douglas Thor, Eifu Tomita, Alfredo Torres, M. Mert Torunbalci, Abeer Vaishnav, Justin Vargas, Sergey Vdovichev, Guifre Vidal, Catherine Vollgraff Heidweiller, Steven Waltman, Jonathan Waltz, Shannon X. Wang, Brayden Ware, Travis Weidel, Theodore White, Kristi Wong, Bryan W. K. Woo, Maddy Woodson, Cheng Xing, Z. Jamie Yao, Ping Yeh, Bicheng Ying, Juhwan Yoo, Noureldin Yosri, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, Sergio Boixo, Julian Kelly, Vadim Smelyanskiy, Hartmut Neven, Dave Bacon, Zijun Chen, Paul V. Klimov, Pedram Roushan, Charles Neill, Yu Chen, and Alexis Morvan, "Demonstration of dynamic surface codes", Nature Physics 21 12, 1994 (2025).

[24] 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).

[25] 尹华磊 Yin Hualei, 卢旭阳 Lu Xuyang, 张庆航 Zhang Qinghang, and 陈增兵 Chen Zengbing, " 基于融合的量子计算研究进展（特邀）", Laser & Optoelectronics Progress 62 11, 1127003 (2025).

[26] Ryota Nakai and Hayato Goto, "Subsystem many-hypercube codes: High-rate concatenated codes with low-weight syndrome measurements", Physical Review Applied 25 1, 014032 (2026).

[27] Derek Khu, Andrew Tanggara, Chao Jin, and Kishor Bharti, "Contextuality of Quantum Error-Correcting Codes", PRX Quantum 7 1, 010319 (2026).

[28] Peter-Jan H.S. Derks, Alex Townsend-Teague, Jens Eisert, Markus S. Kesselring, Oscar Higgott, and Benjamin J. Brown, "Dynamical codes for hardware with noisy readouts", Quantum 10, 2176 (2026).

[29] Julio C. Magdalena de la Fuente, Josias Old, Alex Townsend-Teague, Manuel Rispler, Jens Eisert, and Markus Müller, "XYZ Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime", PRX Quantum 6 1, 010360 (2025).

[30] Ian Hesner, Bence Hetényi, and James R. Wootton, "Using detector likelihood for benchmarking quantum error correction", Physical Review A 111 5, 052452 (2025).

[31] Linnea Grans-Samuelsson, David Aasen, and Parsa Bonderson, "Fault-tolerant pairwise measurement-based code on eight qubits", Physical Review A 112 4, 042413 (2025).

[32] Gözde Üstün and Simon J. Devitt, "Comparing schemes for creating qudit graph states from 16- and 128-dimensional Hilbert space using donors in silicon", Physical Review Research 8 1, 013343 (2026).

[33] Kwok Ho Wan, Zhenghao Zhong, and Ainhoa Zapirain, "Simulating magic state cultivation with few Clifford terms", Quantum 10, 2134 (2026).

[34] Michael Liaofan Liu, Nathanan Tantivasadakarn, and Victor V. Albert, "Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma", Quantum 8, 1403 (2024).

[35] J. A. Muniz, D. Crow, H. Kim, J. M. Kindem, W. B. Cairncross, A. Ryou, T. C. Bohdanowicz, C.-A. Chen, Y. Ji, A. M. W. Jones, E. Megidish, C. Nishiguchi, M. Urbanek, L. Wadleigh, T. Wilkason, D. Aasen, K. Barnes, J. M. Bello-Rivas, I. Bloomfield, G. Booth, A. Brown, M. O. Brown, K. Cassella, G. Cowan, J. Epstein, M. Feldkamp, C. Griger, Y. Hassan, A. Heinz, E. Halperin, T. Hofler, F. Hummel, M. Jaffe, E. Kapit, K. Kotru, J. Lauigan, J. Marjanovic, M. Meredith, M. McDonald, R. Morshead, S. Narayanaswami, K. A. Pawlak, K. L. Pudenz, D. Rodríguez Pérez, P. Sabharwal, J. Simon, A. Smull, M. Sorensen, D. T. Stack, M. Stone, L. Taneja, R. J. M. van de Veerdonk, Z. Vendeiro, R. T. Weverka, K. White, T.-Y. Wu, X. Xie, E. Zalys-Geller, X. Zhang, J. King, B. J. Bloom, and M. A. Norcia, "Repeated Ancilla Reuse for Logical Computation on a Neutral Atom Quantum Computer", Physical Review X 15 4, 041040 (2025).

[36] M. Sohaib Alam and Eleanor Rieffel, "Dynamical logical qubits in the Bacon-Shor code", Physical Review A 112 2, 022436 (2025).

[37] Harashta Tatimma Larasati and Byung-Soo Choi, "Hybrid Circuit and Measurement-Based Quantum Computing Architecture With Encoded-Fusion-Assisted Teleportation", IEEE Access 14, 68662 (2026).

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

[39] 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).

[40] Markus S. Kesselring, Julio C. Magdalena de la Fuente, Felix Thomsen, Jens Eisert, Stephen D. Bartlett, and Benjamin J. Brown, "Anyon Condensation and the Color Code", PRX Quantum 5 1, 010342 (2024).

[41] Matt McEwen, Dave Bacon, and Craig Gidney, "Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics", Quantum 7, 1172 (2023).

[42] 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).

[43] Tyler D. Ellison, Joseph Sullivan, and Arpit Dua, "Floquet codes with a twist", arXiv:2306.08027, (2023).

[44] Arpit Dua, Nathanan Tantivasadakarn, Joseph Sullivan, and Tyler D. Ellison, "Engineering 3D Floquet Codes by Rewinding", PRX Quantum 5 2, 020305 (2024).

[45] 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).

[46] Héctor Bombín, Mihir Pant, Sam Roberts, and Karthik I. Seetharam, "Fault-Tolerant Postselection for Low-Overhead Magic State Preparation", PRX Quantum 5 1, 010302 (2024).

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

[48] Hector Bombin, Chris Dawson, Terry Farrelly, Yehua Liu, Naomi Nickerson, Mihir Pant, Fernando Pastawski, and Sam Roberts, "Fault-tolerant complexes", arXiv:2308.07844, (2023).

[49] Andrew Tanggara, Mile Gu, and Kishor Bharti, "Simple Construction of Qudit Floquet Codes on a Family of Lattices", arXiv:2410.02022, (2024).

[50] Alex Townsend-Teague, Julio Magdalena de la Fuente, and Markus Kesselring, "Floquetifying the Colour Code", arXiv:2307.11136, (2023).

[51] Andreas Bauer, "Topological error correcting processes from fixed-point path integrals", Quantum 8, 1288 (2024).

[52] Craig Gidney and Thiago Bergamaschi, "A Constant Rate Quantum Computer on a Line", arXiv:2502.16132, (2025).

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

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

[55] Andrew Tanggara, Mile Gu, and Kishor Bharti, "Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction", arXiv:2405.17567, (2024).

[56] Keller Blackwell and Jeongwan Haah, "The code distance of Floquet codes", arXiv:2510.05549, (2025).

[57] Peter-Jan H. S. Derks, Alex Townsend-Teague, Jens Eisert, Markus S. Kesselring, Oscar Higgott, and Benjamin J. Brown, "Dynamical codes for hardware with noisy readouts", arXiv:2505.07658, (2025).

[58] Andreas Bauer, "Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases", arXiv:2403.12119, (2024).

[59] 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).

[60] Giovanni de Felice, Boldizsár Poór, Lia Yeh, and William Cashman, "Fusion and flow: formal protocols to reliably build photonic graph states", arXiv:2409.13541, (2024).

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

[62] Nicolas Delfosse, Adam Paetznick, Jeongwan Haah, and Matthew B. Hastings, "Splitting decoders for correcting hypergraph faults", arXiv:2309.15354, (2023).

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

[64] Kwok Ho Wan and Zhenghao Zhong, "Pauli web of the $|Y\rangle$ state surface code injection", arXiv:2501.15566, (2025).

[65] Dhruv Bhatnagar, Matthew Steinberg, David Elkouss, Carmen G. Almudever, and Sebastian Feld, "Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes", arXiv:2305.00784, (2023).

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

[67] Ashot Avanesov, Alexander Shurinov, Ivan Dyakonov, and Stanislav Straupe, "Building a fusion-based quantum computer using teleported gates", Quantum 9, 1762 (2025).

[68] Nicolas Delfosse and Adam Paetznick, "Simulation of noisy Clifford circuits without fault propagation", arXiv:2309.15345, (2023).

[69] Kwok Ho Wan, "Iteratively decoded magic state distillation", arXiv:2410.17992, (2024).

[70] Bhakti Shah, Willam Spencer, Laura Zielinski, Ben Caldwell, Adrian Lehmann, and Robert Rand, "ViCAR: Visualizing Categories with Automated Rewriting in Coq", arXiv:2404.08163, (2024).

[71] Maximilian Rüsch, Aleks Kissinger, and Benjamin Rodatz, "Completeness for Fault Equivalence of Clifford ZX Diagrams", arXiv:2510.08477, (2025).

[72] Timo Hillmann, "Bosonic quantum computing with near-term devices and beyond", arXiv:2512.15063, (2025).

[73] Kwok Ho Wan and Zhenghao Zhong, "Pauli webs spun by transversal $|Y\rangle$ state initialisation", arXiv:2502.00957, (2025).

[74] 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", arXiv:2512.00141, (2025).

[75] Aleks Kissinger and John van de Wetering, "ZX-Flow: A Flexible Criterion for Deterministic Computation with ZX-Diagrams", arXiv:2603.09580, (2026).

[76] Kwok Ho Wan, H. C. W. Price, and Qing Yao, "Holographic codes seen through ZX-calculus", arXiv:2601.04467, (2026).

[77] Maximilian Schweikart, Linnea Grans-Samuelsson, Aleks Kissinger, and Benjamin Rodatz, "Preserving MWPM-Decodability in Fault-Equivalent Rewrites", arXiv:2603.19522, (2026).

[78] 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).

[79] Robert I. Booth and Cole Comfort, "Denotational semantics for stabiliser quantum programs", arXiv:2511.22734, (2025).

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

[81] 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).

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