A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

Daniel Litinski

doi:10.22331/q-2019-03-05-128

A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

Daniel Litinski

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

Published:	2019-03-05, volume 3, page 128
Eprint:	arXiv:1808.02892v3
Doi:	https://doi.org/10.22331/q-2019-03-05-128
Citation:	Quantum 3, 128 (2019).

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Abstract

Given a quantum gate circuit, how does one execute it in a fault-tolerant architecture with as little overhead as possible? In this paper, we discuss strategies for surface-code quantum computing on small, intermediate and large scales. They are strategies for space-time trade-offs, going from slow computations using few qubits to fast computations using many qubits. Our schemes are based on surface-code patches, which not only feature a low space cost compared to other surface-code schemes, but are also conceptually simple~--~simple enough that they can be described as a tile-based game with a small set of rules. Therefore, no knowledge of quantum error correction is necessary to understand the schemes in this paper, but only the concepts of qubits and measurements.

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Popular summary

Useful, classically intractable quantum computations can be very long, potentially consisting of billions of quantum gates. Some of these computations use as few as 100 qubits, but these need to be logical error-corrected qubits, rather than physical qubits, as the coherence times of currently available physical qubits are many orders of magnitude shorter than the execution times of these computations. With logical qubits, the set of available operation is determined by the error-correcting code, independently of the underlying physical hardware. For full-scale quantum computations on hundreds of logical qubits (i.e., potentially hundreds of thousands of physical qubits), it is useful to have a framework that describes logical qubits and operations without keeping track of the details of the physical hardware and error-correction protocols. This paper introduces such a framework with a focus on surface codes, but can also be applied to other topological codes. It uses lattice surgery to describe all logical operations via the easy-to-understand concepts of qubits and measurements, avoiding anyons and topological braiding diagrams. Using this framework, a complete full-scale quantum computer is constructed, consisting of qubit blocks that perform magic state distillation and blocks that consume magic states to advance the computation. Finally, space-time trade-offs are discussed, i.e., how to use more qubits to compute faster.

► BibTeX data

@article{Litinski2019gameofsurfacecodes,
  doi = {10.22331/q-2019-03-05-128},
  url = {https://doi.org/10.22331/q-2019-03-05-128},
  title = {A {G}ame of {S}urface {C}odes: {L}arge-{S}cale {Q}uantum {C}omputing with {L}attice {S}urgery},
  author = {Litinski, Daniel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {128},
  month = mar,
  year = {2019}
}

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[44] Theodoros Trochatos, Christopher Kang, Andrew Wang, Frederic T. Chong, and Jakub Szefer, 2026 IEEE International Symposium on High Performance Computer Architecture (HPCA) 1 (2026) ISBN:979-8-3315-9302-5.

[45] 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.

[46] Jasper J. Postema and Servaas J. J. M. F. Kokkelmans, "Geometrical approach to logical qubit fidelities of neutral atom Calderbank–Shor–Steane codes", Academia Quantum 2 1(2025).

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[50] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes", PRX Quantum 3 1, 010329 (2022).

[51] Christopher Chamberland and Andrew W. Cross, "Fault-tolerant magic state preparation with flag qubits", Quantum 3, 143 (2019).

[52] Jiaxuan Zhang, Tian-Hao Wei, Xi-Ning Zhuang, Zhao-Yun Chen, Wei-Cheng Kong, Yu-Chun Wu, and Guo-Ping Guo, "Low-Overhead and High-Fidelity Preparation of Logical Non-Clifford States with Multilevel Transversal Injection", Physical Review Letters 136 8, 080603 (2026).

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[57] Avimita Chatterjee, Archisman Ghosh, and Swaroop Ghosh, 2026 27th International Symposium on Quality Electronic Design (ISQED) 1 (2026) ISBN:979-8-3315-8361-3.

[58] Zebo Yang, Maede Zolanvari, and Raj Jain, "A Survey of Important Issues in Quantum Computing and Communications", IEEE Communications Surveys & Tutorials 25 2, 1059 (2023).

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[65] Vivien Vandaele, "Lower T-count with faster algorithms", Quantum 9, 1860 (2025).

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[67] Timothy N. Georges, Marius Bothe, Christoph Sünderhauf, Bjorn K. Berntson, Róbert Izsák, and Aleksei V. Ivanov, "Quantum simulations of chemistry in first quantization with any basis set", npj Quantum Information 11 1, 55 (2025).

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[71] Fang Zhang, Xing Zhu, Rui Chao, Cupjin Huang, Linghang Kong, Guoyang Chen, Dawei Ding, Haishan Feng, Yihuai Gao, Xiaotong Ni, Liwei Qiu, Zhe Wei, Yueming Yang, Yang Zhao, Yaoyun Shi, Weifeng Zhang, Peng Zhou, and Jianxin Chen, "A Classical Architecture for Digital Quantum Computers", ACM Transactions on Quantum Computing 5 1, 1 (2024).

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[80] Andrew Patterson and Leigh Lapworth, "Measurement schemes for quantum linear equation solvers", Quantum Science and Technology 10 2, 025037 (2025).

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[83] Yifei Wang and Yingfei Gu, "Non-Clifford quantum gate teleportation with generalized lattice surgery", Physical Review A 113 6, 062410 (2026).

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[85] Yuri Alexeev, Maximilian Amsler, Marco Antonio Barroca, Sanzio Bassini, Torey Battelle, Daan Camps, David Casanova, Young Jay Choi, Frederic T. Chong, Charles Chung, Christopher Codella, Antonio D. Córcoles, James Cruise, Alberto Di Meglio, Ivan Duran, Thomas Eckl, Sophia Economou, Stephan Eidenbenz, Bruce Elmegreen, Clyde Fare, Ismael Faro, Cristina Sanz Fernández, Rodrigo Neumann Barros Ferreira, Keisuke Fuji, Bryce Fuller, Laura Gagliardi, Giulia Galli, Jennifer R. Glick, Isacco Gobbi, Pranav Gokhale, Salvador de la Puente Gonzalez, Johannes Greiner, Bill Gropp, Michele Grossi, Emanuel Gull, Burns Healy, Matthew R. Hermes, Benchen Huang, Travis S. Humble, Nobuyasu Ito, Artur F. Izmaylov, Ali Javadi-Abhari, Douglas Jennewein, Shantenu Jha, Liang Jiang, Barbara Jones, Wibe Albert de Jong, Petar Jurcevic, William Kirby, Stefan Kister, Masahiro Kitagawa, Joel Klassen, Katherine Klymko, Kwangwon Koh, Masaaki Kondo, Dog̃a Murat Kürkçüog̃lu, Krzysztof Kurowski, Teodoro Laino, Ryan Landfield, Matt Leininger, Vicente Leyton-Ortega, Ang Li, Meifeng Lin, Junyu Liu, Nicolas Lorente, Andre Luckow, Simon Martiel, Francisco Martin-Fernandez, Margaret Martonosi, Claire Marvinney, Arcesio Castaneda Medina, Dirk Merten, Antonio Mezzacapo, Kristel Michielsen, Abhishek Mitra, Tushar Mittal, Kyungsun Moon, Joel Moore, Sarah Mostame, Mario Motta, Young-Hye Na, Yunseong Nam, Prineha Narang, Yu-ya Ohnishi, Daniele Ottaviani, Matthew Otten, Scott Pakin, Vincent R. Pascuzzi, Edwin Pednault, Tomasz Piontek, Jed Pitera, Patrick Rall, Gokul Subramanian Ravi, Niall Robertson, Matteo A.C. Rossi, Piotr Rydlichowski, Hoon Ryu, Georgy Samsonidze, Mitsuhisa Sato, Nishant Saurabh, Vidushi Sharma, Kunal Sharma, Soyoung Shin, George Slessman, Mathias Steiner, Iskandar Sitdikov, In-Saeng Suh, Eric D. Switzer, Wei Tang, Joel Thompson, Synge Todo, Minh C. Tran, Dimitar Trenev, Christian Trott, Huan-Hsin Tseng, Norm M. Tubman, Esin Tureci, David García Valiñas, Sofia Vallecorsa, Christopher Wever, Konrad Wojciechowski, Xiaodi Wu, Shinjae Yoo, Nobuyuki Yoshioka, Victor Wen-zhe Yu, Seiji Yunoki, Sergiy Zhuk, and Dmitry Zubarev, "Quantum-centric supercomputing for materials science: A perspective on challenges and future directions", Future Generation Computer Systems 160, 666 (2024).

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[100] Yutaro Akahoshi, Jun Fujisaki, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 420 (2024) ISBN:979-8-3315-4137-8.

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[106] Tobias V. Forster, Nils Quetschlich, Mathias Soeken, and Robert Wille, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 681 (2025) ISBN:979-8-3315-5736-2.

[107] Hammam Qassim, Joel J. Wallman, and Joseph Emerson, "Clifford recompilation for faster classical simulation of quantum circuits", Quantum 3, 170 (2019).

[108] Spiro Gicev, Lloyd C. L. Hollenberg, and Muhammad Usman, "Quantum computer error structure probed by quantum error correction syndrome measurements", Physical Review Research 6 4, 043249 (2024).

[109] Zhu Sun and Bálint Koczor, "Space and time cost of continuous rotations in surface codes", Quantum Science and Technology 11 2, 025046 (2026).

[110] 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.

[111] Shuchen Zhu, Aarthi Sundaram, and Guang Hao Low, "Unified architecture for quantum lookup tables", Physical Review Research 7 4, 043230 (2025).

[112] Jungmin Cho, Hyeonseong Jeong, Junpyo Kim, Junhyuk Choi, Juwon Hong, and Jangwoo Kim, Proceedings of the 31st ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 2 59 (2026) ISBN:9798400723599.

[113] Ivan B. Djordjevic, Quantum Communication, Quantum Networks, and Quantum Sensing 563 (2022) ISBN:9780128229422.

[114] Takuya Kasamura, Junichiro Kadomoto, and Hidetsugu Irie, 2025 IEEE 43rd International Conference on Computer Design (ICCD) 823 (2025) ISBN:979-8-3315-0346-8.

[115] Junichiro Kadomoto, Ren Aoyama, and Kazutoshi Kobayashi, 2025 IEEE International Symposium on Circuits and Systems (ISCAS) 1 (2025) ISBN:979-8-3503-5683-0.

[116] Qian Xu, Pei Zeng, Daohong Xu, and Liang Jiang, "Fault-Tolerant Operation of Bosonic Qubits with Discrete-Variable Ancillae", Physical Review X 14 3, 031016 (2024).

[117] Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian, "On the Realistic Worst-Case Analysis of Quantum Arithmetic Circuits", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[118] Zi-Han Chen, Ming-Cheng Chen, Chao-Yang Lu, and Jian-Wei Pan, "Efficient Magic State Cultivation on RP2", PRX Quantum 7 1, 010315 (2026).

[119] Shi WANG, Yan ZHENG, Jie HOU, Yongjin YE, Yang JI, and Yongzheng WU, "Systematical study of effects of chip parameters and control waveforms on fidelity of CZ gate", Acta Physica Sinica 74 15, 150302 (2025).

[120] Athena Caesura, Cristian L. Cortes, William Pol, Sukin Sim, Mark Steudtner, Gian-Luca R. Anselmetti, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, Michael Streif, and Christofer S. Tautermann, "Faster Quantum Chemistry Simulations on a Quantum Computer with Improved Tensor Factorization and Active Volume Compilation", PRX Quantum 6 3, 030337 (2025).

[121] Sam McArdle, "Learning from Physics Experiments with Quantum Computers: Applications in Muon Spectroscopy", PRX Quantum 2 2, 020349 (2021).

[122] Alon Kukliansky and Brad Lackey, "Quantum Circuit Tensors and Enumerators With Applications to Quantum Fault Tolerance", IEEE Transactions on Information Theory 71 6, 4406 (2025).

[123] Matthew Amy and Joseph Lunderville, "Linear and Non-linear Relational Analyses for Quantum Program Optimization", Proceedings of the ACM on Programming Languages 9 POPL, 1072 (2025).

[124] Sapthagiri Miriyala and Venkata Ramireddy Chirra, "From Quantum Algorithm Primitives to Design Patterns: A Unified Framework for NISQ and Fault‐Tolerant Regimes", Advanced Quantum Technologies 9 6, e70310 (2026).

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[130] Mingfei Yu, Alessandro Tempia Calvino, Mathias Soeken, and Giovanni De Micheli, Proceedings of the 30th Asia and South Pacific Design Automation Conference 930 (2025) ISBN:9798400706356.

[131] Hezi Zhang, Yiran Xu, Haotian Hu, Keyi Yin, Hassan Shapourian, Jiapeng Zhao, Ramana Rao Kompella, Reza Nejabati, and Yufei Ding, Proceedings of the 52nd Annual International Symposium on Computer Architecture 1449 (2025) ISBN:9798400712616.

[132] Pei Zeng, Jinzhao Sun, Liang Jiang, and Qi Zhao, "Simple and High-Precision Hamiltonian Simulation by Compensating Trotter Error with Linear Combination of Unitary Operations", PRX Quantum 6 1, 010359 (2025).

[133] Shota Kanasugi, Yuichiro Hidaka, Yuya O. Nakagawa, Shoichiro Tsutsui, Norifumi Matsumoto, Kazunori Maruyama, Hirotaka Oshima, and Shintaro Sato, "Subspace-based local compilation of variational quantum circuits for large-scale quantum many-body simulation", Physical Review Research 7 2, 023298 (2025).

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[135] Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia, and Earl T. Campbell, "Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes", Physical Review X 13 3, 031007 (2023).

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[137] Seok-Hyung Lee, Felix Thomsen, Nicholas Fazio, Benjamin J. Brown, and Stephen D. Bartlett, "Low-Overhead Magic State Distillation with Color Codes", PRX Quantum 6 3, 030317 (2025).

[138] William J. Huggins and Jarrod R. McClean, "Accelerating Quantum Algorithms with Precomputation", Quantum 8, 1264 (2024).

[139] Jason D. Chadwick, Mariesa H. Teo, Joshua Viszlai, Willers Yang, and Frederic T. Chong, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 658 (2025) ISBN:979-8-3315-5736-2.

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[141] Simon Burton and Dan Browne, "Limitations on Transversal Gates for Hypergraph Product Codes", IEEE Transactions on Information Theory 68 3, 1772 (2022).

[142] Soronzonbold Otgonbaatar and Dieter Kranzlmüller, "Exploiting the Quantum Advantage for Satellite Image Processing: Review and Assessment", IEEE Transactions on Quantum Engineering 5, 1 (2024).

[143] Ankith Konda, 2025 13th International Conference on Intelligent Systems and Embedded Design (ISED) 936 (2025) ISBN:979-8-3315-6726-2.

[144] S.N. Saadatmand, Tyler L. Wilson, Mark J. Hodson, Mark Field, Simon J. Devitt, Madhav Krishnan Vijayan, Alan Robertson, Thinh P. Le, Jannis Ruh, Alexandru Paler, Arshpreet Singh Maan, Ioana Moflic, Athena Caesura, and Josh Y. Mutus, "Superconducting qubits in the millions: The potential and limitations of modularity", Physical Review Applied 25 6, 064038 (2026).

[145] Áron Márton and János K. Asbóth, "Optimal number of stabilizer measurement rounds in an idling surface code patch", Quantum 9, 1767 (2025).

[146] Namitha Liyanage, Yue Wu, Emmet Houghton, and Lin Zhong, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 1148 (2025) ISBN:979-8-3315-5736-2.

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[150] 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).

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