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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[64] Yutaka Hirano and Keisuke Fujii, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 670 (2025) ISBN:979-8-3315-5736-2.

[65] Yuri Alexeev, Marwa H. Farag, Taylor L. Patti, Mark E. Wolf, Natalia Ares, Alán Aspuru-Guzik, Simon C. Benjamin, Zhenyu Cai, Shuxiang Cao, Christopher Chamberland, Zohim Chandani, Federico Fedele, Ikko Hamamura, Nicholas Harrigan, Jin-Sung Kim, Elica Kyoseva, Justin G. Lietz, Tom Lubowe, Alexander McCaskey, Roger G. Melko, Kouhei Nakaji, Alberto Peruzzo, Pooja Rao, Bruno Schmitt, Sam Stanwyck, Norm M. Tubman, Hanrui Wang, and Timothy Costa, "Artificial intelligence for quantum computing", Nature Communications 16 1, 10829 (2025).

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

[69] 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.

[70] 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.

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[72] 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).

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[75] 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).

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[102] Nick S. Blunt, Joan Camps, Ophelia Crawford, Róbert Izsák, Sebastian Leontica, Arjun Mirani, Alexandra E. Moylett, Sam A. Scivier, Christoph Sünderhauf, Patrick Schopf, Jacob M. Taylor, and Nicole Holzmann, "Perspective on the Current State-of-the-Art of Quantum Computing for Drug Discovery Applications", Journal of Chemical Theory and Computation 18 12, 7001 (2022).

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[111] James Mills, Adithya Sireesh, Dominik Leichtle, Joschka Roffe, and Elham Kashefi, "Logical Accreditation: A Framework for Efficient Certification of Fault-Tolerant Computations", PRX Quantum 7 3, 033053 (2026).

[112] Raghu Kulkarni, "Inter-Sheet Joint Pauli Measurements on the FCC Lattice: A Cross-Block Fault-Tolerant Primitive at K = 4 Active Connectivity", Quantum Reports 8 3, 98 (2026).

[113] Jiaxuan Zhang, Zhao-Yun Chen, Yun-Jie Wang, Bin-Han Lu, Hai-Feng Zhang, Jia-Ning Li, Peng Duan, Yu-Chun Wu, and Guo-Ping Guo, "Demonstrating a universal logical gate set in error-detecting surface codes on a superconducting quantum processor", npj Quantum Information 11 1, 177 (2025).

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[118] Linghang Kong, Fang Zhang, and Jianxin Chen, "Benchmarking fault-tolerant quantum computing hardware via QLOPS", ACM Transactions on Quantum Computing 7 2, 1 (2026).

[119] Earl Campbell, Ankur Khurana, and Ashley Montanaro, "Applying quantum algorithms to constraint satisfaction problems", Quantum 3, 167 (2019).

[120] Shilin Huang, Tomas Jochym-O’Connor, and Theodore J. Yoder, "Homomorphic Logical Measurements", PRX Quantum 4 3, 030301 (2023).

[121] Christopher Chamberland and Earl T. Campbell, "Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery", PRX Quantum 3 1, 010331 (2022).

[122] Hengyun Zhou, Casey Duckering, Chen Zhao, Dolev Bluvstein, Madelyn Cain, Aleksander Kubica, Sheng-Tao Wang, and Mikhail D. Lukin, Proceedings of the 52nd Annual International Symposium on Computer Architecture 1432 (2025) ISBN:9798400712616.

[123] Siddhant Singh, Rikiya Kashiwagi, Kazufumi Tanji, Wojciech Roga, Daniel Bhatti, Masahiro Takeoka, and David Elkouss, "Fault-tolerant modular quantum computing with surface codes using single-shot emission-based hardware", Physical Review Applied 26 2, 024005 (2026).

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[125] Shashikiran Venkatesha and Ranjani Parthasarathi, "Survey on Redundancy Based-Fault tolerance methods for Processors and Hardware accelerators - Trends in Quantum Computing, Heterogeneous Systems and Reliability", ACM Computing Surveys 56 11, 1 (2024).

[126] Salonik Resch and Ulya Karpuzcu, "On Variable Strength Quantum ECC", IEEE Computer Architecture Letters 21 2, 93 (2022).

[127] John Blue, Harshil Avlani, Zhiyang He, Liu Ziyin, and Isaac L. Chuang, "Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes", Quantum 10, 2149 (2026).

[128] Alex Townsend-Teague, Julio Magdalena de la Fuente, and Markus Kesselring, "Floquetifying the Colour Code", Electronic Proceedings in Theoretical Computer Science 384, 265 (2023).

[129] 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).

[130] Ali Passian, Gilles Buchs, Christopher M. Seck, Alberto M. Marino, and Nicholas A. Peters, "The Concept of a Quantum Edge Simulator: Edge Computing and Sensing in the Quantum Era", Sensors 23 1, 115 (2022).

[131] Josu Etxezarreta Martinez, Paul Schnabl, Javier Oliva del Moral, Reza Dastbasteh, Pedro M. Crespo, and Ruben M. Otxoa, "Leveraging biased noise for more efficient quantum error correction at the circuit level with two-level qubits", Physical Review Applied 25 1, 014021 (2026).

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[133] Tim Chan and Simon C. Benjamin, "Actis: A Strictly Local Union–Find Decoder", Quantum 7, 1183 (2023).

[134] Mark L. Turner, Earl T. Campbell, Ophelia Crawford, Neil I. Gillespie, and Joan Camps, "Scalable Decoding Protocols for Fast Transversal Logic in the Surface Code", PRX Quantum 7 1, 010320 (2026).

[135] Samira Sayedsalehi, Nader Bagherzadeh, Alberto A. Del Barrio, Guillermo Botella, and Ratko Pilipović, "Developing and Analyzing the Defect-Based Surface Codes Using Optimization Algorithms", Quantum Reports 7 2, 25 (2025).

[136] Haruki Mitarai, Yukihiro Tadokoro, and Hiroya Tanaka, "Active interference suppression in frequency-division-multiplexed quantum gates via off-resonant microwave tones", Physical Review Research 8 3, 033376 (2026).

[137] Kaiwen Zhou, Liqiang Lu, Debin Xiang, Chenning Tao, Anbang Wu, Jingwen Leng, Fangxin Liu, Mingshuai Chen, and Jianwei Yin, Proceedings of the 58th IEEE/ACM International Symposium on Microarchitecture 719 (2025) ISBN:9798400715730.

[138] Songqinghao Yang and Prakash Murali, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 428 (2025) ISBN:979-8-3315-5736-2.

[139] Ioana Moflic and Alexandru Paler, "On the constant depth implementation of Pauli exponentials", npj Quantum Information 12 1, 82 (2026).

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[144] Mingzheng Zhu, Hao Fu, Haishan Song, Jun Wu, Chi Zhang, Wei Xie, and Xiangyang Li, "Ecmas+: Efficient Circuit Mapping and Scheduling for Surface Code Encoded Circuit on Quantum Cloud Platform", ACM Transactions on Architecture and Code Optimization 22 3, 1 (2025).

[145] Archisman Ghosh, Avimita Chatterjee, and Swaroop Ghosh, 2026 31st Asia and South Pacific Design Automation Conference (ASP-DAC) 198 (2026) ISBN:979-8-3315-9123-6.

[146] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer, "Trading T gates for dirty qubits in state preparation and unitary synthesis", Quantum 8, 1375 (2024).

[147] Michael Garn and Angus Kan, "Quantum Resource Estimates for Computing Binary Elliptic Curve Discrete Logarithms", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[148] Junpyo Kim, Jungmin Cho, Hyeonseong Jeong, Dongmoon Min, Junhyuk Choi, Juwon Hong, and Jangwoo Kim, Proceedings of the 58th IEEE/ACM International Symposium on Microarchitecture 547 (2025) ISBN:9798400715730.

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[150] Amanda Xu, Abtin Molavi, Swamit Tannu, and Aws Albarghouthi, Proceedings of the 30th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 1 777 (2025) ISBN:9798400706981.

[151] György P. Gehér, Campbell McLauchlan, Earl T. Campbell, Alexandra E. Moylett, and Ophelia Crawford, "Error-corrected Hadamard gate simulated at the circuit level", Quantum 8, 1394 (2024).

[152] Jason Gavriel, Daniel Herr, Alexis Shaw, Michael J. Bremner, Alexandru Paler, and Simon J. Devitt, "Transversal injection for direct encoding of ancilla states for non-Clifford gates using stabilizer codes", Physical Review Research 5 3, 033019 (2023).

[153] Alexandru Paler and Austin G. Fowler, "Pipelined correlated minimum weight perfect matching of the surface code", Quantum 7, 1205 (2023).

[154] Michael Vasmer and Aleksander Kubica, "Morphing Quantum Codes", PRX Quantum 3 3, 030319 (2022).

[155] Guoding Liu, Xingjian Zhang, and Xiongfeng Ma, "Classically Replaceable Operations", Quantum 6, 845 (2022).

[156] Riki Toshio, Yutaro Akahoshi, Jun Fujisaki, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii, "Practical Quantum Advantage on Partially Fault-Tolerant Quantum Computer", Physical Review X 15 2, 021057 (2025).

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