Optimization of the surface code design for Majorana-based qubits

Rui Chao; Michael E. Beverland; Nicolas Delfosse; Jeongwan Haah

doi:10.22331/q-2020-10-28-352

Optimization of the surface code design for Majorana-based qubits

Rui Chao¹, Michael E. Beverland², Nicolas Delfosse², and Jeongwan Haah²

¹University of Southern California, Los Angeles, CA, USA
²Microsoft Quantum and Microsoft Research, Redmond, WA, USA

Published:	2020-10-28, volume 4, page 352
Eprint:	arXiv:2007.00307v2
Doi:	https://doi.org/10.22331/q-2020-10-28-352
Citation:	Quantum 4, 352 (2020).

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Abstract

The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.

Here, we present surface code error-correction schemes using $\textit{only}$ Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.

Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.

Featured image: The windmill layout for implementing the distance-5 surface code with measurement-based qubits. Black nodes are data qubits; at the center of each X-type or Z-type plaquette, there is a green or yellow ancilla qubit, respectively. The measurement of a plaquette will require two-qubit measurements between the data qubits and the center ancilla (black edges), and between the center ancilla and its neighbor ancilla (blue edge).

► BibTeX data

@article{Chao2020optimizationof,
  doi = {10.22331/q-2020-10-28-352},
  url = {https://doi.org/10.22331/q-2020-10-28-352},
  title = {Optimization of the surface code design for {M}ajorana-based qubits},
  author = {Chao, Rui and Beverland, Michael E. and Delfosse, Nicolas and Haah, Jeongwan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {352},
  month = oct,
  year = {2020}
}

► References

[1] A. Y. Kitaev, Ann. Phys. 303, 2 (2003), quant-ph/9707021.
https://doi.org/10.1016/S0003-4916(02)00018-0
arXiv:quant-ph/9707021

[2] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002), quant-ph/0110143.
https://doi.org/10.1063/1.1499754
arXiv:quant-ph/0110143

[3] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Phys. Rev. A 86, 032324 (2012), 1208.0928.
https://doi.org/10.1103/PhysRevA.86.032324
arXiv:1208.0928

[4] R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007), quant-ph/0610082.
https://doi.org/10.1103/PhysRevLett.98.190504
arXiv:quant-ph/0610082

[5] A. G. Fowler, A. M. Stephens, and P. Groszkowski, Phys. Rev. A 80, 052312 (2009), 0803.0272.
https://doi.org/10.1103/PhysRevA.80.052312
arXiv:0803.0272

[6] N. Delfosse and G. Zémor, Physical Review Research 2, 033042 (2020), 1703.01517.
https://doi.org/10.1103/PhysRevResearch.2.033042
arXiv:1703.01517

[7] N. Delfosse and N. H. Nickerson, 2017, 1709.06218.
arXiv:1709.06218

[8] T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, Phys. Rev. B 95, 235305 (2017), 1610.05289.
https://doi.org/10.1103/PhysRevB.95.235305
arXiv:1610.05289

[9] C. Knapp, M. Beverland, D. I. Pikulin, and T. Karzig, Quantum 2, 88 (2018), 1806.01275.
https://doi.org/10.22331/q-2018-09-03-88
arXiv:1806.01275

[10] Y. Li, Physical Review Letters 117, 120403 (2016), 1512.05089.
https://doi.org/10.1103/PhysRevLett.117.120403
arXiv:1512.05089

[11] S. Plugge, L. Landau, E. Sela, A. Altland, K. Flensberg, and R. Egger, Phys. Rev. B 94, 174514 (2016), 1606.08408.
https://doi.org/10.1103/PhysRevB.94.174514
arXiv:1606.08408

[12] D. Litinski, M. S. Kesselring, J. Eisert, and F. von Oppen, Phys. Rev. X 7, 031048 (2017), 1704.01589.
https://doi.org/10.1103/PhysRevX.7.031048
arXiv:1704.01589

[13] A. G. Fowler, D. S. Wang, and L. C. L. Hollenberg, Quant. Info. Comput. 11, 8 (2011), 1004.0255.
https://doi.org/10.26421/QIC11.1-2
arXiv:1004.0255

[14] M. Newman, L. A. de Castro, and K. R. Brown, Quantum 4, 295 (2020), 1909.11817.
https://doi.org/10.22331/q-2020-07-13-295
arXiv:1909.11817

[15] A. G. Fowler, 2013, 1310.0863.
arXiv:1310.0863

[16] N. Delfosse and J.-P. Tillich, in 2014 IEEE Int. Symp. Info. (IEEE, 2014) pp. 1071–1075, 1401.6975.
https://doi.org/10.1109/ISIT.2014.6874997
arXiv:1401.6975

[17] S. Huang and K. R. Brown, Phys. Rev. A 101, 042312 (2020), 1911.11317.
https://doi.org/10.1103/PhysRevA.101.042312
arXiv:1911.11317

[18] S. Huang, M. Newman, and K. R. Brown, Phys. Rev. A 102, 012419 (2020), 2004.04693.
https://doi.org/10.1103/PhysRevA.102.012419
arXiv:2004.04693

[19] J. MacWilliams, Bell Syst. Tech. J. 42, 79 (1963).
https://doi.org/10.1002/j.1538-7305.1963.tb04003.x

[20] S. Bravyi and A. Vargo, Phys. Rev. A 88, 062308 (2013), 1308.6270.
https://doi.org/10.1103/PhysRevA.88.062308
arXiv:1308.6270

Cited by

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

[2] Craig Gidney, "Stim: a fast stabilizer circuit simulator", Quantum 5, 497 (2021).

[3] Qian Xu, Guo Zheng, Yu-Xin Wang, Peter Zoller, Aashish A. Clerk, and Liang Jiang, "Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits", npj Quantum Information 9 1, 78 (2023).

[4] Kaavya Sahay, Jahan Claes, and Shruti Puri, "Tailoring Fusion-Based Error Correction for High Thresholds to Biased Fusion Failures", Physical Review Letters 131 12, 120604 (2023).

[5] Maryam Mudassar, Alexander Schuckert, and Daniel Gottesman, "Fault-Tolerant Operations in Majorana-Based Quantum Codes: Gates, Measurements, and High-Rate Constructions", PRX Quantum 7 2, 020315 (2026).

[6] Mojtaba Hassani and Zahra Faraei, "Interplay of tilt and axion fields in topological superconductors: Anisotropy in the Meissner effect", Physical Review B 112 13, 134524 (2025).

[7] Jeongwan Haah and Matthew B. Hastings, "Boundaries for the Honeycomb Code", Quantum 6, 693 (2022).

[8] Aleksander Kubica and Michael Vasmer, "Single-shot quantum error correction with the three-dimensional subsystem toric code", Nature Communications 13 1, 6272 (2022).

[9] Theerapat Tansuwannont and Debbie Leung, "Achieving Fault Tolerance on Capped Color Codes with Few Ancillas", PRX Quantum 3 3, 030322 (2022).

[10] Edward H. Chen, Theodore J. Yoder, Youngseok Kim, Neereja Sundaresan, Srikanth Srinivasan, Muyuan Li, Antonio D. Córcoles, Andrew W. Cross, and Maika Takita, "Calibrated Decoders for Experimental Quantum Error Correction", Physical Review Letters 128 11, 110504 (2022).

[11] Craig Gidney, Michael Newman, and Matt McEwen, "Benchmarking the Planar Honeycomb Code", Quantum 6, 813 (2022).

[12] Kaavya Sahay and Benjamin J. Brown, "Decoder for the Triangular Color Code by Matching on a Möbius Strip", PRX Quantum 3 1, 010310 (2022).

[13] Michael E. Beverland, Aleksander Kubica, and Krysta M. Svore, "Cost of Universality: A Comparative Study of the Overhead of State Distillation and Code Switching with Color Codes", PRX Quantum 2 2, 020341 (2021).

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

[15] Oscar Higgott and Craig Gidney, "Sparse Blossom: correcting a million errors per core second with minimum-weight matching", Quantum 9, 1600 (2025).

[16] Dongxiao Quan, Chensong Liu, Xiaojie Lv, and Changxing Pei, "Implementation of Fault-Tolerant Encoding Circuit Based on Stabilizer Implementation and “Flag” Bits in Steane Code", Entropy 24 8, 1107 (2022).

[17] Min Ye and Nicolas Delfosse, "Quantum error correction for long chains of trapped ions", Quantum 9, 1920 (2025).

[18] Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton, "A Fault-Tolerant Honeycomb Memory", Quantum 5, 605 (2021).

[19] Linnea Grans-Samuelsson, Ryan V. Mishmash, David Aasen, Christina Knapp, Bela Bauer, Brad Lackey, Marcus P. da Silva, and Parsa Bonderson, "Improved Pairwise Measurement-Based Surface Code", Quantum 8, 1429 (2024).

[20] 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).

[21] 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).

[22] "Basic Theory and Implementations of Quantum Error Correction", University of Denver (2025).

[23] Ali Fahimniya, Hossein Dehghani, Kishor Bharti, Sheryl Mathew, Alicia J. Kollár, Alexey V. Gorshkov, and Michael J. Gullans, "Fault-tolerant hyperbolic Floquet quantum error correcting codes", Quantum 9, 1849 (2025).

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

[25] Qian Xu, Nam Mannucci, Alireza Seif, Aleksander Kubica, Steven T. Flammia, and Liang Jiang, "Tailored XZZX codes for biased noise", Physical Review Research 5 1, 013035 (2023).

[26] Xiu-Bo Chen, Li-Yun Zhao, Gang Xu, Xing-Bo Pan, Si-Yi Chen, Zhen-Wen Cheng, and Yi-Xian Yang, "Low-overhead fault-tolerant error correction scheme based on quantum stabilizer codes", Chinese Physics B 31 4, 040305 (2022).

[27] Shouzhen Gu, Alex Retzker, and Aleksander Kubica, "Fault-tolerant quantum architectures based on erasure qubits", Physical Review Research 7 1, 013249 (2025).

[28] 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 1, 010310 (2023).

[29] Oscar Higgott and Nikolas P. Breuckmann, "Constructions and Performance of Hyperbolic and Semi-Hyperbolic Floquet Codes", PRX Quantum 5 4, 040327 (2024).

[30] Shouzhen Gu, Yotam Vaknin, Alex Retzker, and Aleksander Kubica, "Optimizing Quantum Error-Correction Protocols with Erasure Qubits", PRX Quantum 6 4, 040354 (2025).

[31] Michael Beverland, Vadym Kliuchnikov, and Eddie Schoute, "Surface Code Compilation via Edge-Disjoint Paths", PRX Quantum 3 2, 020342 (2022).

[32] David Aasen, Jeongwan Haah, Zhi Li, and Roger S. K. Mong, "Measurement Quantum Cellular Automata and Anomalies in Floquet Codes", arXiv:2304.01277, (2023).

[33] 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).

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

[35] Satoshi Yoshida, Craig Gidney, Matt McEwen, and Adam Zalcman, "Low Depth Color Code Circuits with CXSWAP gate", arXiv:2510.00370, (2025).

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