Improved Pairwise Measurement-Based Surface Code

Linnea Grans-Samuelsson; Ryan V. Mishmash; David Aasen; Christina Knapp; Bela Bauer; Brad Lackey; Marcus P. da Silva; Parsa Bonderson

doi:10.22331/q-2024-08-02-1429

Improved Pairwise Measurement-Based Surface Code

Linnea Grans-Samuelsson¹, Ryan V. Mishmash¹, David Aasen¹, Christina Knapp¹, Bela Bauer¹, Brad Lackey², Marcus P. da Silva², and Parsa Bonderson¹

¹Microsoft Station Q, Santa Barbara, California 93106-6105 USA
²Microsoft Quantum, Redmond, Washington 98052, USA

Published:	2024-08-02, volume 8, page 1429
Eprint:	arXiv:2310.12981v2
Doi:	https://doi.org/10.22331/q-2024-08-02-1429
Citation:	Quantum 8, 1429 (2024).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We devise a new realization of the surface code on a rectangular lattice of qubits utilizing single-qubit and nearest-neighbor two-qubit Pauli measurements and three auxiliary qubits per plaquette. This realization gains substantial advantages over prior pairwise measurement-based realizations of the surface code. It has a short operation period of 4 steps and our performance analysis for a standard circuit noise model yields a high fault-tolerance threshold of approximately $0.66\% $. The syndrome extraction circuits avoid bidirectional hook errors, so we can achieve full code distance by choosing appropriate boundary conditions. We also construct variants of the syndrome extraction circuits that entirely prevent hook errors, at the cost of larger circuit depth. This achieves full distance regardless of boundary conditions, with only a modest decrease in the threshold. Furthermore, we propose an efficient strategy for dealing with dead components (qubits and measurements) in our surface code realization, which can be adopted more generally for other surface code realizations. This new surface code realization is highly optimized for Majorana-based hardware, accounting for constraints imposed by layouts and the implementation of measurements, making it competitive with the recently proposed Floquet codes.

► BibTeX data

@article{GransSamuelsson2024improvedpairwise,
  doi = {10.22331/q-2024-08-02-1429},
  url = {https://doi.org/10.22331/q-2024-08-02-1429},
  title = {Improved {P}airwise {M}easurement-{B}ased {S}urface {C}ode},
  author = {Grans-Samuelsson, Linnea and Mishmash, Ryan V. and Aasen, David and Knapp, Christina and Bauer, Bela and Lackey, Brad and Silva, Marcus P. da and Bonderson, Parsa},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1429},
  month = aug,
  year = {2024}
}

► References

[1] Alexei Yu. Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of Physics 303, 2–30 (2003). arXiv:quant-ph/9707021.
https://doi.org/10.1016/S0003-4916(02)00018-0
arXiv:quant-ph/9707021

[2] Sergey B. Bravyi and Alexei Yu. Kitaev. ``Quantum codes on a lattice with boundary'' (1998). arXiv:quant-ph/9811052.
arXiv:quant-ph/9811052

[3] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. Journal of Mathematical Physics 43, 4452–4505 (2002). arXiv:quant-ph/0110143.
https://doi.org/10.1063/1.1499754
arXiv:quant-ph/0110143

[4] Michael E. Beverland, Prakash Murali, Matthias Troyer, Krysta M. Svore, Torsten Hoefler, Vadym Kliuchnikov, Guang Hao Low, Mathias Soeken, Aarthi Sundaram, and Alexander Vaschillo. ``Assessing requirements to scale to practical quantum advantage'' (2022). arXiv:2211.07629.
arXiv:2211.07629

[5] 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). arXiv:2202.11829.
https://doi.org/10.1103/PRXQuantum.4.010310
arXiv:2202.11829

[6] Torsten Karzig, Christina Knapp, Roman M. Lutchyn, Parsa Bonderson, Matthew B. Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, Yuval Oreg, Charles M. Marcus, and Michael H. Freedman. ``Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes''. Phys. Rev. B 95, 235305 (2017). arXiv:1610.05289.
https://doi.org/10.1103/PhysRevB.95.235305
arXiv:1610.05289

[7] Alan Tran, Alex Bocharov, Bela Bauer, and Parsa Bonderson. ``Optimizing Clifford gate generation for measurement-only topological quantum computation with Majorana zero modes''. SciPost Phys. 8, 091 (2020). arXiv:1909.03002.
https://doi.org/10.21468/SciPostPhys.8.6.091
arXiv:1909.03002

[8] Rui Chao, Michael E. Beverland, Nicolas Delfosse, and Jeongwan Haah. ``Optimization of the surface code design for Majorana-based qubits''. Quantum 4, 352 (2020). arXiv:2007.00307.
https://doi.org/10.22331/q-2020-10-28-352
arXiv:2007.00307

[9] Craig Gidney. ``A Pair Measurement Surface Code on Pentagons'' (2022). arXiv:2206.12780.
https://doi.org/10.22331/q-2023-10-25-1156
arXiv:2206.12780

[10] Matthew B. Hastings and Jeongwan Haah. ``Dynamically Generated Logical Qubits''. Quantum 5, 564 (2021). arXiv:2107.02194.
https://doi.org/10.22331/q-2021-10-19-564
arXiv:2107.02194

[11] Jeongwan Haah and Matthew B. Hastings. ``Boundaries for the Honeycomb Code''. Quantum 6, 693 (2022). arXiv:2110.09545.
https://doi.org/10.22331/q-2022-04-21-693
arXiv:2110.09545

[12] Craig Gidney, Michael Newman, and Matt McEwen. ``Benchmarking the Planar Honeycomb Code''. Quantum 6, 813 (2022). arXiv:2202.11845.
https://doi.org/10.22331/q-2022-09-21-813
arXiv:2202.11845

[13] John van de Wetering. ``ZX-calculus for the working quantum computer scientist'' (2020). arXiv:2012.13966.
arXiv:2012.13966

[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). arXiv:2112.12160.
https://doi.org/10.1103/PRXQuantum.4.020303
arXiv:2112.12160

[15] Héctor Bombín and Miguel A. Martin-Delgado. ``Optimal resources for topological two-dimensional stabilizer codes: Comparative study''. Phys. Rev. A 76, 012305 (2007). arXiv:quant-ph/0703272.
https://doi.org/10.1103/PhysRevA.76.012305
arXiv:quant-ph/0703272

[16] Yu Tomita and Krysta M. Svore. ``Low-distance surface codes under realistic quantum noise''. Phys. Rev. A 90, 062320 (2014). arXiv:1404.3747.
https://doi.org/10.1103/PhysRevA.90.062320
arXiv:1404.3747

[17] James M. Auger, Hussain Anwar, Mercedes Gimeno-Segovia, Thomas M. Stace, and Dan E. Browne. ``Fault-tolerance thresholds for the surface code with fabrication errors''. Phys. Rev. A 96, 042316 (2017). arXiv:1706.04912.
https://doi.org/10.1103/PhysRevA.96.042316
arXiv:1706.04912

[18] Christopher Chamberland and Michael E. Beverland. ``Flag fault-tolerant error correction with arbitrary distance codes''. Quantum 2, 53 (2018). arXiv:1708.02246.
https://doi.org/10.22331/q-2018-02-08-53
arXiv:1708.02246

[19] Craig Gidney. ``Stim: a fast stabilizer circuit simulator''. Quantum 5, 497 (2021). arXiv:2103.02202.
https://doi.org/10.22331/q-2021-07-06-497
arXiv:2103.02202

[20] Thomas M. Stace, Sean D. Barrett, and Andrew C. Doherty. ``Thresholds for Topological Codes in the Presence of Loss''. Phys. Rev. Lett. 102, 200501 (2009). arXiv:0904.3556.
https://doi.org/10.1103/PhysRevLett.102.200501
arXiv:0904.3556

[21] Armands Strikis, Simon C. Benjamin, and Benjamin J. Brown. ``Quantum Computing is Scalable on a Planar Array of Qubits with Fabrication Defects''. Phys. Rev. Applied 19, 064081 (2023). arXiv:2111.06432.
https://doi.org/10.1103/PhysRevApplied.19.064081
arXiv:2111.06432

[22] Oscar Higgott and Craig Gidney. ``Sparse Blossom: correcting a million errors per core second with minimum-weight matching'' (2023). arXiv:2303.15933.
arXiv:2303.15933

[23] Nicolas Delfosse and Adam Paetznick. ``Spacetime codes of Clifford circuits'' (2023). arXiv:2304.05943.
arXiv:2304.05943

[24] Nicolas Delfosse, Adam Paetznick, Jeongwan Haah, and Mathew B. Hastings. ``Splitting decoders for correcting hypergraph faults'' (2023). arXiv:2309.15354.
arXiv:2309.15354

[25] Richard M. Karp. ``Reducibility among combinatorial problems''. Pages 85–103. Springer US. Boston, MA (1972).
https://doi.org/10.1007/978-1-4684-2001-2_9

[26] Christopher A. Pattison, Michael E. Beverland, Marcus P. da Silva, and Nicolas Delfosse. ``Improved quantum error correction using soft information'' (2021). arXiv:2107.13589.
arXiv:2107.13589

[27] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Phys. Rev. A 86, 032324 (2012). arXiv:1208.0928.
https://doi.org/10.1103/PhysRevA.86.032324
arXiv:1208.0928

Cited by

[1] Tyler LeBlond, Peter Groszkowski, Justin G. Lietz, Christopher M. Seck, and Ryan S. Bennink, "Logical error rates for the surface code under a mixed coherent and stochastic circuit-level noise model inspired by trapped ions", Physical Review Research 7 4, 043184 (2025).

[2] Tyler LeBlond and Ryan S. Bennink, "Quantum Resource Comparison for Two Leading Surface Code Lattice Surgery Approaches", Quantum 10, 2187 (2026).

[3] Morteza Aghaee, Alejandro Alcaraz Ramirez, Zulfi Alam, Rizwan Ali, Mariusz Andrzejczuk, Andrey Antipov, Mikhail Astafev, Amin Barzegar, Bela Bauer, Jonathan Becker, Umesh Kumar Bhaskar, Alex Bocharov, Srini Boddapati, David Bohn, Jouri Bommer, Leo Bourdet, Arnaud Bousquet, Samuel Boutin, Lucas Casparis, Benjamin J. Chapman, Sohail Chatoor, Anna Wulff Christensen, Cassandra Chua, Patrick Codd, William Cole, Paul Cooper, Fabiano Corsetti, Ajuan Cui, Paolo Dalpasso, Juan Pablo Dehollain, Gijs de Lange, Michiel de Moor, Andreas Ekefjärd, Tareq El Dandachi, Juan Carlos Estrada Saldaña, Saeed Fallahi, Luca Galletti, Geoff Gardner, Deshan Govender, Flavio Griggio, Ruben Grigoryan, Sebastian Grijalva, Sergei Gronin, Jan Gukelberger, Marzie Hamdast, Firas Hamze, Esben Bork Hansen, Sebastian Heedt, Zahra Heidarnia, Jesús Herranz Zamorano, Samantha Ho, Laurens Holgaard, John Hornibrook, Jinnapat Indrapiromkul, Henrik Ingerslev, Lovro Ivancevic, Thomas Jensen, Jaspreet Jhoja, Jeffrey Jones, Konstantin V. Kalashnikov, Ray Kallaher, Rachpon Kalra, Farhad Karimi, Torsten Karzig, Evelyn King, Maren Elisabeth Kloster, Christina Knapp, Dariusz Kocon, Jonne V. Koski, Pasi Kostamo, Mahesh Kumar, Tom Laeven, Thorvald Larsen, Jason Lee, Kyunghoon Lee, Grant Leum, Kongyi Li, Tyler Lindemann, Matthew Looij, Julie Love, Marijn Lucas, Roman Lutchyn, Morten Hannibal Madsen, Nash Madulid, Albert Malmros, Michael Manfra, Devashish Mantri, Signe Brynold Markussen, Esteban Martinez, Marco Mattila, Robert McNeil, Antonio B. Mei, Ryan V. Mishmash, Gopakumar Mohandas, Christian Mollgaard, Trevor Morgan, George Moussa, Chetan Nayak, Jens Hedegaard Nielsen, Jens Munk Nielsen, William Hvidtfelt Padkar Nielsen, Bas Nijholt, Mike Nystrom, Eoin O’Farrell, Thomas Ohki, Keita Otani, Brian Paquelet Wütz, Sebastian Pauka, Karl Petersson, Luca Petit, Dima Pikulin, Guen Prawiroatmodjo, Frank Preiss, Eduardo Puchol Morejon, Mohana Rajpalke, Craig Ranta, Katrine Rasmussen, David Razmadze, Outi Reentila, David J. Reilly, Yuan Ren, Ken Reneris, Richard Rouse, Ivan Sadovskyy, Lauri Sainiemi, Irene Sanlorenzo, Emma Schmidgall, Cristina Sfiligoj, Mustafeez Bashir Shah, Kevin Simoes, Shilpi Singh, Sarat Sinha, Thomas Soerensen, Patrick Sohr, Tomas Stankevic, Lieuwe Stek, Eric Stuppard, Henri Suominen, Judith Suter, Sam Teicher, Nivetha Thiyagarajah, Raj Tholapi, Mason Thomas, Emily Toomey, Josh Tracy, Michelle Turley, Shivendra Upadhyay, Ivan Urban, Kevin Van Hoogdalem, David J. Van Woerkom, Dmitrii V. Viazmitinov, Dominik Vogel, John Watson, Alex Webster, Joseph Weston, Georg W. Winkler, Di Xu, Chung Kai Yang, Emrah Yucelen, Roland Zeisel, Guoji Zheng, and Justin Zilke, "Interferometric single-shot parity measurement in InAs–Al hybrid devices", Nature 638 8051, 651 (2025).

[4] 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).

[5] Campbell McLauchlan, György P. Gehér, and Alexandra E. Moylett, "Accommodating Fabrication Defects on Floquet Codes with Minimal Hardware Requirements", Quantum 8, 1562 (2024).

[6] César Benito, Esperanza López, Borja Peropadre, and Alejandro Bermudez, "Comparative study of quantum error correction strategies for the heavy-hexagonal lattice", Quantum 9, 1623 (2025).

[7] Dripto M. Debroy, Matt McEwen, Craig Gidney, Noah Shutty, and Adam Zalcman, "LUCI in the Surface Code with Dropouts", Quantum 9, 1936 (2025).

[8] Hengyun Zhou, Madelyn Cain, and Mikhail D. Lukin, "Opportunities in full-stack design of low-overhead fault-tolerant quantum computation", Nature Computational Science 5 12, 1110 (2025).

[9] Shi Jie Samuel Tan and Lev Stambler, "Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes", Quantum 9, 1897 (2025).

[10] David Aasen, Morteza Aghaee, Zulfi Alam, Mariusz Andrzejczuk, Andrey Antipov, Mikhail Astafev, Lukas Avilovas, Amin Barzegar, Bela Bauer, Jonathan Becker, Juan M. Bello-Rivas, Umesh Bhaskar, Alex Bocharov, Srini Boddapati, David Bohn, Jouri Bommer, Parsa Bonderson, Jan Borovsky, Leo Bourdet, Samuel Boutin, Tom Brown, Gary Campbell, Lucas Casparis, Srivatsa Chakravarthi, Rui Chao, Benjamin J. Chapman, Sohail Chatoor, Anna Wulff Christensen, Patrick Codd, William Cole, Paul Cooper, Fabiano Corsetti, Ajuan Cui, Wim van Dam, Tareq El Dandachi, Sahar Daraeizadeh, Adrian Dumitrascu, Andreas Ekefjärd, Saeed Fallahi, Luca Galletti, Geoff Gardner, Raghu Gatta, Haris Gavranovic, Michael Goulding, Deshan Govender, Flavio Griggio, Ruben Grigoryan, Sebastian Grijalva, Sergei Gronin, Jan Gukelberger, Jeongwan Haah, Marzie Hamdast, Esben Bork Hansen, Matthew Hastings, Sebastian Heedt, Samantha Ho, Justin Hogaboam, Laurens Holgaard, Kevin Van Hoogdalem, Jinnapat Indrapiromkul, Henrik Ingerslev, Lovro Ivancevic, Sarah Jablonski, Thomas Jensen, Jaspreet Jhoja, Jeffrey Jones, Kostya Kalashnikov, Ray Kallaher, Rachpon Kalra, Farhad Karimi, Torsten Karzig, Seth Kimes, Vadym Kliuchnikov, Maren Elisabeth Kloster, Christina Knapp, Derek Knee, Jonne Koski, Pasi Kostamo, Jamie Kuesel, Brad Lackey, Tom Laeven, Jeffrey Lai, Gijs de Lange, Thorvald Larsen, Jason Lee, Kyunghoon Lee, Grant Leum, Kongyi Li, Tyler Lindemann, Marijn Lucas, Roman Lutchyn, Morten Hannibal Madsen, Nash Madulid, Michael Manfra, Signe Brynold Markussen, Esteban Martinez, Marco Mattila, Jake Mattinson, Robert McNeil, Antonio Rodolph Mei, Ryan V. Mishmash, Gopakumar Mohandas, Christian Mollgaard, Michiel de Moor, Trevor Morgan, George Moussa, Anirudh Narla, Chetan Nayak, Jens Hedegaard Nielsen, William Hvidtfelt Padkær Nielsen, Frédéric Nolet, Mike Nystrom, Eoin O’Farrell, Keita Otani, Adam Paetznick, Camille Papon, Andres Paz, Karl Petersson, Luca Petit, Dima Pikulin, Diego Olivier Fernandez Pons, Sam Quinn, Mohana Rajpalke, Alejandro Alcaraz Ramirez, Katrine Rasmussen, David Razmadze, Ben Reichardt, Yuan Ren, Ken Reneris, Roy Riccomini, Ivan Sadovskyy, Lauri Sainiemi, Juan Carlos Estrada Saldaña, Irene Sanlorenzo, Simon Schaal, Emma Schmidgall, Cristina Sfiligoj, Marcus P. da Silva, Shilpi Singh, Sarat Sinha, Mathias Soeken, Patrick Sohr, Tomas Stankevic, Lieuwe Stek, Patrick Strøm-Hansen, Eric Stuppard, Aarthi Sundaram, Henri Suominen, Judith Suter, Satoshi Suzuki, Krysta Svore, Sam Teicher, Nivetha Thiyagarajah, Raj Tholapi, Mason Thomas, Dennis Tom, Emily Toomey, Josh Tracy, Matthias Troyer, Michelle Turley, Matthew D. Turner, Shivendra Upadhyay, Ivan Urban, Alexander Vaschillo, Dmitrii Viazmitinov, Dominik Vogel, Zhenghan Wang, John Watson, Alex Webster, Joseph Weston, Timothy Williamson, Georg W. Winkler, David J. van Woerkom, Brian Paquelet Wütz, Chung Kai Yang, Richard Yu, Emrah Yucelen, Jesús Herranz Zamorano, Roland Zeisel, Guoji Zheng, Justin Zilke, and Andrew Zimmerman, "Blueprint for fault-tolerant quantum computation with topological qubit arrays", Physical Review Research 7 4, 041002 (2025).

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

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

[13] Ryan V. Mishmash, Vadym Kliuchnikov, Juan Bello-Rivas, Adam Paetznick, David Aasen, Christina Knapp, Yue Wu, Bela Bauer, Marcus P. da Silva, and Parsa Bonderson, "Excising dead components in the surface code using minimally invasive alterations: A performance study", arXiv:2508.04786, (2025).

[14] Alon Kukliansky and Brad Lackey, "Quantum Circuit Tensors and Enumerators with Applications to Quantum Fault Tolerance", arXiv:2405.19643, (2024).

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

[16] Benjamin Anker and Dripto M. Debroy, "Optimized Measurement Schedules for the Surface Code with Dropout", arXiv:2512.10871, (2025).

[17] Dominic J. Williamson and Bence Hetényi, "Dynamical quantum codes and logic gates on a lattice with sparse connectivity", arXiv:2510.05225, (2025).

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

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 01:41:31) and SAO/NASA ADS (last updated successfully 2026-08-12 01:41:34). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.