Comparative study of quantum error correction strategies for the heavy-hexagonal lattice

César Benito; Esperanza López; Borja Peropadre; Alejandro Bermudez

doi:10.22331/q-2025-02-06-1623

Comparative study of quantum error correction strategies for the heavy-hexagonal lattice

César Benito¹, Esperanza López¹, Borja Peropadre², and Alejandro Bermudez^3,1

¹Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain
²IBM Quantum, IBM Research, Cambridge, MA 02142, USA
³Currently on sabbatical at Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom

Published:	2025-02-06, volume 9, page 1623
Editor:	Narayanan Rengaswamy
Eprint:	arXiv:2402.02185v2
Doi:	https://doi.org/10.22331/q-2025-02-06-1623
Citation:	Quantum 9, 1623 (2025).

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

Abstract

Topological quantum error correction is a milestone in the scaling roadmap of quantum computers, which targets circuits with trillions of gates that would allow running quantum algorithms for real-world problems. The square-lattice surface code has become the workhorse to address this challenge, as it poses milder requirements on current devices both in terms of required error rates and small local connectivities. In some platforms, however, the connectivities are kept even lower in order to minimise gate errors at the hardware level, which limits the error correcting codes that can be directly implemented on them. In this work, we make a comparative study of possible strategies to overcome this limitation for the heavy-hexagonal lattice, the architecture of current IBM superconducting quantum computers. We explore two complementary strategies: the search for an efficient embedding of the surface code into the heavy-hexagonal lattice, as well as the use of codes whose connectivity requirements are naturally tailored to this architecture, such as subsystem-type and Floquet codes. Using noise models of increased complexity, we assess the performance of these strategies for IBM devices in terms of their error thresholds and qubit footprints. An optimized SWAP-based embedding of the surface code is found to be the most promising strategy towards a near-term demonstration of quantum error correction advantage.

Featured image: QEC-advantage footprint for IBM devices: Number of physical qubits required to preserve a logical qubit for $d$ QEC rounds with a logical error rate lower than $10^{-4}$, considering two different IBM-devices with their corresponding PBCL noise models. We plot the QEC footprint as a function of the ratio $\chi$ between improved error and the current calibrated error rate. The surface code with SWAPs has the best behaviour, which nevertheless is not enough to show QEC advantage without a reduction of existing noise.

Popular summary

Quantum error correction is a key milestone in the roadmap for quantum computers, as it enables the execution of complex quantum algorithms for real-world problems. Although achieving this goal is still beyond the capabilities of state-of-the-art quantum processors, it remains valuable to run small-scale demonstrations of error-corrected logical qubits on near-term devices. In this work, we compared different strategies for embedding error-correcting codes into the architecture of IBM Quantum systems, which utilize a heavy-hexagonal qubit layout. Our findings show that the well-known surface code can be embedded within the heavy-hexagonal lattice by using SWAP gates with minimal overhead. This approach represents one of the most promising strategies for a near-term demonstration of quantum error correction advantage on these devices.

► BibTeX data

@article{Benito2025comparativestudyof,
  doi = {10.22331/q-2025-02-06-1623},
  url = {https://doi.org/10.22331/q-2025-02-06-1623},
  title = {Comparative study of quantum error correction strategies for the heavy-hexagonal lattice},
  author = {Benito, C{\'{e}}sar and L{\'{o}}pez, Esperanza and Peropadre, Borja and Bermudez, Alejandro},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1623},
  month = feb,
  year = {2025}
}

► References

[1] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''. Cambridge University Press. (2000).
https://doi.org/10.1017/CBO9780511976667

[2] Ashley Montanaro. ``Quantum algorithms: an overview''. npj Quantum Information 2, 15023 (2016).
https://doi.org/10.1038/npjqi.2015.23

[3] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão. ``Quantum algorithms: A survey of applications and end-to-end complexities'' (2023). arXiv:2310.03011.
arXiv:2310.03011

[4] W. G. Unruh. ``Maintaining coherence in quantum computers''. Phys. Rev. A 51, 992–997 (1995).
https://doi.org/10.1103/PhysRevA.51.992

[5] Frank. Arute et al. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019).
https://doi.org/10.1038/s41586-019-1666-5

[6] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, Peng Hu, Xiao-Yan Yang, Wei-Jun Zhang, Hao Li, Yuxuan Li, Xiao Jiang, Lin Gan, Guangwen Yang, Lixing You, Zhen Wang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan. ``Quantum computational advantage using photons''. Science 370, 1460–1463 (2020).
https://doi.org/10.1126/science.abe8770

[7] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta. ``Error mitigation for short-depth quantum circuits''. Phys. Rev. Lett. 119, 180509 (2017).
https://doi.org/10.1103/PhysRevLett.119.180509

[8] Paul D. Nation, Hwajung Kang, Neereja Sundaresan, and Jay M. Gambetta. ``Scalable mitigation of measurement errors on quantum computers''. PRX Quantum 2, 040326 (2021).
https://doi.org/10.1103/PRXQuantum.2.040326

[9] Ewout van den Berg, Zlatko K. Minev, Abhinav Kandala, and Kristan Temme. ``Probabilistic error cancellation with sparse pauli–lindblad models on noisy quantum processors''. Nature Physics 19, 1116–1121 (2023).
https://doi.org/10.1038/s41567-023-02042-2

[10] Zhenyu Cai, Ryan Babbush, Simon C. Benjamin, Suguru Endo, William J. Huggins, Ying Li, Jarrod R. McClean, and Thomas E. O'Brien. ``Quantum error mitigation''. Rev. Mod. Phys. 95, 045005 (2023).
https://doi.org/10.1103/RevModPhys.95.045005

[11] Lorenza Viola, Emanuel Knill, and Seth Lloyd. ``Dynamical decoupling of open quantum systems''. Phys. Rev. Lett. 82, 2417–2421 (1999).
https://doi.org/10.1103/PhysRevLett.82.2417

[12] K. Khodjasteh and D. A. Lidar. ``Fault-tolerant quantum dynamical decoupling''. Phys. Rev. Lett. 95, 180501 (2005).
https://doi.org/10.1103/PhysRevLett.95.180501

[13] Kaveh Khodjasteh and Lorenza Viola. ``Dynamically error-corrected gates for universal quantum computation''. Phys. Rev. Lett. 102, 080501 (2009).
https://doi.org/10.1103/PhysRevLett.102.080501

[14] Daniel A. Lidar. ``Review of decoherence-free subspaces, noiseless subsystems, and dynamical decoupling''. Pages 295–354. John Wiley & Sons, Ltd. (2014).
https://doi.org/10.1002/9781118742631.ch11

[15] John Preskill. ``Quantum Computing in the NISQ era and beyond''. Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[16] Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout van den Berg, Sami Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, and Abhinav Kandala. ``Evidence for the utility of quantum computing before fault tolerance''. Nature 618, 500–505 (2023).
https://doi.org/10.1038/s41586-023-06096-3

[17] A. R. Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Phys. Rev. A 54, 1098–1105 (1996).
https://doi.org/10.1103/PhysRevA.54.1098

[18] A. M. Steane. ``Error correcting codes in quantum theory''. Phys. Rev. Lett. 77, 793–797 (1996).
https://doi.org/10.1103/PhysRevLett.77.793

[19] Barbara M. Terhal. ``Quantum error correction for quantum memories''. Rev. Mod. Phys. 87, 307–346 (2015).
https://doi.org/10.1103/RevModPhys.87.307

[20] D. Aharonov and M. Ben-Or. ``Fault-tolerant quantum computation with constant error''. SIAM J. Comput. 38, 1207 (1998).
https://doi.org/10.1137/S0097539799359385

[21] Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek. ``Resilient quantum computation: error models and thresholds''. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365–384 (1998).
https://doi.org/10.1098/rspa.1998.0166

[22] Colin D. Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M. Sage. ``Trapped-ion quantum computing: Progress and challenges''. Applied Physics Reviews 6, 021314 (2019).
https://doi.org/10.1063/1.5088164

[23] M. Morgado and S. Whitlock. ``Quantum simulation and computing with Rydberg-interacting qubits''. AVS Quantum Science 3, 023501 (2021).
https://doi.org/10.1116/5.0036562

[24] G Wendin. ``Quantum information processing with superconducting circuits: a review''. Reports on Progress in Physics 80, 106001 (2017).
https://doi.org/10.1088/1361-6633/aa7e1a

[25] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O'Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis. ``Superconducting quantum circuits at the surface code threshold for fault tolerance''. Nature 508, 500–503 (2014).
https://doi.org/10.1038/nature13171

[26] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas. ``High-fidelity quantum logic gates using trapped-ion hyperfine qubits''. Phys. Rev. Lett. 117, 060504 (2016).
https://doi.org/10.1103/PhysRevLett.117.060504

[27] Simon J. Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H. Li, Alexandra A. Geim, Tout T. Wang, Nishad Maskara, Harry Levine, Giulia Semeghini, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. ``High-fidelity parallel entangling gates on a neutral-atom quantum computer''. Nature 622, 268–272 (2023).
https://doi.org/10.1038/s41586-023-06481-y

[28] Craig R. Clark, Holly N. Tinkey, Brian C. Sawyer, Adam M. Meier, Karl A. Burkhardt, Christopher M. Seck, Christopher M. Shappert, Nicholas D. Guise, Curtis E. Volin, Spencer D. Fallek, Harley T. Hayden, Wade G. Rellergert, and Kenton R. Brown. ``High-fidelity bell-state preparation with $^{40}{\mathrm{ca}}^{+}$ optical qubits''. Phys. Rev. Lett. 127, 130505 (2021).
https://doi.org/10.1103/PhysRevLett.127.130505

[29] Leon Ding, Max Hays, Youngkyu Sung, Bharath Kannan, Junyoung An, Agustin Di Paolo, Amir H. Karamlou, Thomas M. Hazard, Kate Azar, David K. Kim, Bethany M. Niedzielski, Alexander Melville, Mollie E. Schwartz, Jonilyn L. Yoder, Terry P. Orlando, Simon Gustavsson, Jeffrey A. Grover, Kyle Serniak, and William D. Oliver. ``High-fidelity, frequency-flexible two-qubit fluxonium gates with a transmon coupler''. Phys. Rev. X 13, 031035 (2023).
https://doi.org/10.1103/PhysRevX.13.031035

[30] Helin Zhang, Chunyang Ding, D.K. Weiss, Ziwen Huang, Yuwei Ma, Charles Guinn, Sara Sussman, Sai Pavan Chitta, Danyang Chen, Andrew A. Houck, Jens Koch, and David I. Schuster. ``Tunable inductive coupler for high-fidelity gates between fluxonium qubits''. PRX Quantum 5, 020326 (2024).
https://doi.org/10.1103/PRXQuantum.5.020326

[31] D. Kielpinski, C. Monroe, and D. J. Wineland. ``Architecture for a large-scale ion-trap quantum computer''. Nature 417, 709–711 (2002).
https://doi.org/10.1038/nature00784

[32] V. Kaushal, B. Lekitsch, A. Stahl, J. Hilder, D. Pijn, C. Schmiegelow, A. Bermudez, M. Müller, F. Schmidt-Kaler, and U. Poschinger. ``Shuttling-based trapped-ion quantum information processing''. AVS Quantum Science 2, 014101 (2020).
https://doi.org/10.1116/1.5126186

[33] Jonathan P. Home, David Hanneke, John D. Jost, Jason M. Amini, Dietrich Leibfried, and David J. Wineland. ``Complete methods set for scalable ion trap quantum information processing''. Science 325, 1227–1230 (2009).
https://doi.org/10.1126/science.1177077

[34] J. M. Pino et al. ``Demonstration of the trapped-ion quantum CCD computer architecture''. Nature 592, 209–213 (2021).
https://doi.org/10.1038/s41586-021-03318-4

[35] C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz. ``Realization of real-time fault-tolerant quantum error correction''. Phys. Rev. X 11, 041058 (2021).
https://doi.org/10.1103/PhysRevX.11.041058

[36] J. Hilder, D. Pijn, O. Onishchenko, A. Stahl, M. Orth, B. Lekitsch, A. Rodriguez-Blanco, M. Müller, F. Schmidt-Kaler, and U. G. Poschinger. ``Fault-tolerant parity readout on a shuttling-based trapped-ion quantum computer''. Phys. Rev. X 12, 011032 (2022).
https://doi.org/10.1103/PhysRevX.12.011032

[37] Jérôme Beugnon, Charles Tuchendler, Harold Marion, Alpha Gaëtan, Yevhen Miroshnychenko, Yvan R. P. Sortais, Andrew M. Lance, Matthew P. A. Jones, Gaétan Messin, Antoine Browaeys, and Philippe Grangier. ``Two-dimensional transport and transfer of a single atomic qubit in optical tweezers''. Nature Physics 3, 696–699 (2007).
https://doi.org/10.1038/nphys698

[38] Daniel Barredo, Sylvain de Léséleuc, Vincent Lienhard, Thierry Lahaye, and Antoine Browaeys. ``An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays''. Science 354, 1021–1023 (2016).
https://doi.org/10.1126/science.aah3778

[39] Manuel Endres, Hannes Bernien, Alexander Keesling, Harry Levine, Eric R. Anschuetz, Alexandre Krajenbrink, Crystal Senko, Vladan Vuletic, Markus Greiner, and Mikhail D. Lukin. ``Atom-by-atom assembly of defect-free one-dimensional cold atom arrays''. Science 354, 1024–1027 (2016).
https://doi.org/10.1126/science.aah3752

[40] Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T. Wang, Sepehr Ebadi, Marcin Kalinowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. ``A quantum processor based on coherent transport of entangled atom arrays''. Nature 604, 451–456 (2022).
https://doi.org/10.1038/s41586-022-04592-6

[41] Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, J Pablo Bonilla Ataides, Nishad Maskara, Iris Cong, Xun Gao, Pedro Sales Rodriguez, Thomas Karolyshyn, Giulia Semeghini, Michael J Gullans, Markus Greiner, Vladan Vuletić, and Mikhail D Lukin. ``Logical quantum processor based on reconfigurable atom arrays''. Nature 626, 58–65 (2024).
https://doi.org/10.1038/s41586-023-06927-3

[42] S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K. Wright, and C. Monroe. ``Demonstration of a small programmable quantum computer with atomic qubits''. Nature 536, 63–66 (2016).
https://doi.org/10.1038/nature18648

[43] C. Figgatt, A. Ostrander, N. M. Linke, K. A. Landsman, D. Zhu, D. Maslov, and C. Monroe. ``Parallel entangling operations on a universal ion-trap quantum computer''. Nature 572, 368–372 (2019).
https://doi.org/10.1038/s41586-019-1427-5

[44] I. Pogorelov, T. Feldker, Ch. D. Marciniak, L. Postler, G. Jacob, O. Krieglsteiner, V. Podlesnic, M. Meth, V. Negnevitsky, M. Stadler, B. Höfer, C. Wächter, K. Lakhmanskiy, R. Blatt, P. Schindler, and T. Monz. ``Compact ion-trap quantum computing demonstrator''. PRX Quantum 2, 020343 (2021).
https://doi.org/10.1103/PRXQuantum.2.020343

[45] Lukas Postler, Friederike Butt, Ivan Pogorelov, Christian D. Marciniak, Sascha Heußen, Rainer Blatt, Philipp Schindler, Manuel Rispler, Markus Müller, and Thomas Monz. ``Demonstration of fault-tolerant steane quantum error correction''. PRX Quantum 5, 030326 (2024).
https://doi.org/10.1103/PRXQuantum.5.030326

[46] Shilin Huang, Kenneth R. Brown, and Marko Cetina. ``Comparing shor and steane error correction using the bacon-shor code''. Science Advances 10, eadp2008 (2024).
https://doi.org/10.1126/sciadv.adp2008

[47] D. Rosenberg, D. Kim, R. Das, D. Yost, S. Gustavsson, D. Hover, P. Krantz, A. Melville, L. Racz, G. O. Samach, S. J. Weber, F. Yan, J. L. Yoder, A. J. Kerman, and W. D. Oliver. ``3d integrated superconducting qubits''. npj Quantum Information 3, 42 (2017).
https://doi.org/10.1038/s41534-017-0044-0

[48] Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder. ``High-threshold and low-overhead fault-tolerant quantum memory''. Nature 627, 778–782 (2024).
https://doi.org/10.1038/s41586-024-07107-7

[49] J. Kelly et al. ``State preservation by repetitive error detection in a superconducting quantum circuit''. Nature 519, 66–69 (2015).
https://doi.org/10.1038/nature14270

[50] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V. Smelyanskiy, A. Megrant, B. Chiaro, A. Dunsworth, K. Arya, R. Barends, B. Burkett, Y. Chen, Z. Chen, A. Fowler, B. Foxen, M. Giustina, R. Graff, E. Jeffrey, T. Huang, J. Kelly, P. Klimov, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, H. Neven, and J. M. Martinis. ``A blueprint for demonstrating quantum supremacy with superconducting qubits''. Science 360, 195–199 (2018).
https://doi.org/10.1126/science.aao4309

[51] Erik Lucero, R. Barends, Y. Chen, J. Kelly, M. Mariantoni, A. Megrant, P. O'Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, Y. Yin, A. N. Cleland, and John M. Martinis. ``Computing prime factors with a josephson phase qubit quantum processor''. Nature Physics 8, 719–723 (2012).
https://doi.org/10.1038/nphys2385

[52] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta. ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets''. Nature 549, 242–246 (2017).
https://doi.org/10.1038/nature23879

[53] Ken X. Wei, Isaac Lauer, Srikanth Srinivasan, Neereja Sundaresan, Douglas T. McClure, David Toyli, David C. McKay, Jay M. Gambetta, and Sarah Sheldon. ``Verifying multipartite entangled greenberger-horne-zeilinger states via multiple quantum coherences''. Phys. Rev. A 101, 032343 (2020).
https://doi.org/10.1103/PhysRevA.101.032343

[54] Sebastian Krinner, Nathan Lacroix, Ants Remm, Agustin Di Paolo, Elie Genois, Catherine Leroux, Christoph Hellings, Stefania Lazar, Francois Swiadek, Johannes Herrmann, Graham J. Norris, Christian Kraglund Andersen, Markus Müller, Alexandre Blais, Christopher Eichler, and Andreas Wallraff. ``Realizing repeated quantum error correction in a distance-three surface code''. Nature 605, 669–674 (2022).
https://doi.org/10.1038/s41586-022-04566-8

[55] Rajeev Acharya, Igor Aleiner, Richard Allen, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Joao Basso, Andreas Bengtsson, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Dripto M. Debroy, Alexander Del Toro Barba, Sean Demura, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatemi, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Alejandro Grajales Dau, Jonathan A. Gross, Steve Habegger, Michael C. Hamilton, Matthew P. Harrigan, Sean D. Harrington, Oscar Higgott, Jeremy Hilton, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Paul V. Klimov, Andrey R. Klots, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny Lee, Brian J. Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Fionn D. Malone, Jeffrey Marshall, Orion Martin, Jarrod R. McClean, Trevor McCourt, Matt McEwen, Anthony Megrant, Bernardo Meurer Costa, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Alexis Morvan, Emily Mount, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Thomas E. O'Brien, Alex Opremcak, John Platt, Andre Petukhov, Rebecca Potter, Leonid P. Pryadko, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Michael J. Shearn, Aaron Shorter, Vladimir Shvarts, Jindra Skruzny, Vadim Smelyanskiy, W. Clarke Smith, George Sterling, Doug Strain, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White, Cheng Xing, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, and Google Quantum AI. ``Suppressing quantum errors by scaling a surface code logical qubit''. Nature 614, 676–681 (2023).
https://doi.org/10.1038/s41586-022-05434-1

[56] A.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

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

[58] 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

[59] 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).
https://doi.org/10.1103/PhysRevA.86.032324

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

[61] Jay M. Gambetta, Jerry M. Chow, and Matthias Steffen. ``Building logical qubits in a superconducting quantum computing system''. npj Quantum Information 3, 2 (2017).
https://doi.org/10.1038/s41534-016-0004-0

[62] Daniel Gottesman. ``Stabilizer codes and quantum error correction'' (1997). arXiv:quant-ph/9705052.
arXiv:quant-ph/9705052

[63] Ashley M. Stephens. ``Fault-tolerant thresholds for quantum error correction with the surface code''. Phys. Rev. A 89, 022321 (2014).
https://doi.org/10.1103/PhysRevA.89.022321

[64] Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski. ``High-threshold universal quantum computation on the surface code''. Phys. Rev. A 80, 052312 (2009).
https://doi.org/10.1103/PhysRevA.80.052312

[65] D. S. Wang, A. G. Fowler, A. M. Stephens, and L. C. L. Hollenberg. ``Threshold error rates for the toric and planar codes''. Quantum Info. Comput. 10, 456–469 (2010).
https://doi.org/10.5555/2011362.2011368

[66] Panos Aliferis, Daniel Gottesman, and John Preskill. ``Quantum accuracy threshold for concatenated distance-3 codes''. Quantum Info. Comput. 6, 97–165 (2006). arXiv:quant-ph/0504218.
arXiv:quant-ph/0504218

[67] Krysta M. Svore, David P. Divincenzo, and Barbara M. Terhal. ``Noise threshold for a fault-tolerant two-dimensional lattice architecture''. Quantum Info. Comput. 7, 297–318 (2007).
https://doi.org/10.5555/2011725.2011727

[68] L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. ``Demonstration of two-qubit algorithms with a superconducting quantum processor''. Nature 460, 240–244 (2009).
https://doi.org/10.1038/nature08121

[69] Youwei Zhao, Yangsen Ye, He-Liang Huang, Yiming Zhang, Dachao Wu, Huijie Guan, Qingling Zhu, Zuolin Wei, Tan He, Sirui Cao, Fusheng Chen, Tung-Hsun Chung, Hui Deng, Daojin Fan, Ming Gong, Cheng Guo, Shaojun Guo, Lianchen Han, Na Li, Shaowei Li, Yuan Li, Futian Liang, Jin Lin, Haoran Qian, Hao Rong, Hong Su, Lihua Sun, Shiyu Wang, Yulin Wu, Yu Xu, Chong Ying, Jiale Yu, Chen Zha, Kaili Zhang, Yong-Heng Huo, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, and Jian-Wei Pan. ``Realization of an error-correcting surface code with superconducting qubits''. Phys. Rev. Lett. 129, 030501 (2022).
https://doi.org/10.1103/PhysRevLett.129.030501

[70] Yangsen Ye, Tan He, He-Liang Huang, Zuolin Wei, Yiming Zhang, Youwei Zhao, Dachao Wu, Qingling Zhu, Huijie Guan, Sirui Cao, Fusheng Chen, Tung-Hsun Chung, Hui Deng, Daojin Fan, Ming Gong, Cheng Guo, Shaojun Guo, Lianchen Han, Na Li, Shaowei Li, Yuan Li, Futian Liang, Jin Lin, Haoran Qian, Hao Rong, Hong Su, Shiyu Wang, Yulin Wu, Yu Xu, Chong Ying, Jiale Yu, Chen Zha, Kaili Zhang, Yong-Heng Huo, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, and Jian-Wei Pan. ``Logical magic state preparation with fidelity beyond the distillation threshold on a superconducting quantum processor''. Phys. Rev. Lett. 131, 210603 (2023).
https://doi.org/10.1103/PhysRevLett.131.210603

[71] Riddhi S. Gupta, Neereja Sundaresan, Thomas Alexander, Christopher J. Wood, Seth T. Merkel, Michael B. Healy, Marius Hillenbrand, Tomas Jochym-O'Connor, James R. Wootton, Theodore J. Yoder, Andrew W. Cross, Maika Takita, and Benjamin J. Brown. ``Encoding a magic state with beyond break-even fidelity''. Nature 625, 259–263 (2024).
https://doi.org/10.1038/s41586-023-06846-3

[72] Jerry M. Chow, A. D. Córcoles, Jay M. Gambetta, Chad Rigetti, B. R. Johnson, John A. Smolin, J. R. Rozen, George A. Keefe, Mary B. Rothwell, Mark B. Ketchen, and M. Steffen. ``Simple all-microwave entangling gate for fixed-frequency superconducting qubits''. Phys. Rev. Lett. 107, 080502 (2011).
https://doi.org/10.1103/PhysRevLett.107.080502

[73] Sarah Sheldon, Easwar Magesan, Jerry M. Chow, and Jay M. Gambetta. ``Procedure for systematically tuning up cross-talk in the cross-resonance gate''. Phys. Rev. A 93, 060302 (2016).
https://doi.org/10.1103/PhysRevA.93.060302

[74] M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. ``Realization of three-qubit quantum error correction with superconducting circuits''. Nature 482, 382–385 (2012).
https://doi.org/10.1038/nature10786

[75] Jerry M. Chow et al. ``Implementing a strand of a scalable fault-tolerant quantum computing fabric''. Nature Communications 5, 4015 EP (2014).
https://doi.org/10.1038/ncomms5015

[76] A. D. Córcoles, Easwar Magesan, Srikanth J. Srinivasan, Andrew W. Cross, M. Steffen, Jay M. Gambetta, and Jerry M. Chow. ``Demonstration of a quantum error detection code using a square lattice of four superconducting qubits''. Nature Communications 6, 6979 (2015).
https://doi.org/10.1038/ncomms7979

[77] Maika Takita, A. D. Córcoles, Easwar Magesan, Baleegh Abdo, Markus Brink, Andrew Cross, Jerry M. Chow, and Jay M. Gambetta. ``Demonstration of weight-four parity measurements in the surface code architecture''. Phys. Rev. Lett. 117, 210505 (2016).
https://doi.org/10.1103/PhysRevLett.117.210505

[78] Maika Takita, Andrew W. Cross, A. D. Córcoles, Jerry M. Chow, and Jay M. Gambetta. ``Experimental demonstration of fault-tolerant state preparation with superconducting qubits''. Phys. Rev. Lett. 119, 180501 (2017).
https://doi.org/10.1103/PhysRevLett.119.180501

[79] Christopher Chamberland, Guanyu Zhu, Theodore J. Yoder, Jared B. Hertzberg, and Andrew W. Cross. ``Topological and subsystem codes on low-degree graphs with flag qubits''. Phys. Rev. X 10, 011022 (2020).
https://doi.org/10.1103/PhysRevX.10.011022

[80] Rui Chao and Ben W. Reichardt. ``Quantum error correction with only two extra qubits''. Phys. Rev. Lett. 121, 050502 (2018).
https://doi.org/10.1103/PhysRevLett.121.050502

[81] Rui Chao and Ben W. Reichardt. ``Flag fault-tolerant error correction for any stabilizer code''. PRX Quantum 1, 010302 (2020).
https://doi.org/10.1103/PRXQuantum.1.010302

[82] Younghun Kim, Jeongsoo Kang, and Younghun Kwon. ``Design of quantum error correcting code for biased error on heavy-hexagon structure''. Quantum Information Processing 22, 230 (2023).
https://doi.org/10.1007/s11128-023-03979-2

[83] 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

[84] Dave Bacon and Andrea Casaccino. ``Quantum error correcting subsystem codes from two classical linear codes'' (2006). arXiv:quant-ph/0610088.
arXiv:quant-ph/0610088

[85] Panos Aliferis and Andrew W. Cross. ``Subsystem fault tolerance with the bacon-shor code''. Phys. Rev. Lett. 98, 220502 (2007).
https://doi.org/10.1103/PhysRevLett.98.220502

[86] John Napp and John Preskill. ``Optimal bacon-shor codes''. Quantum Inf. Comput. 13, 490–510 (2013). arXiv:1209.0794.
arXiv:1209.0794

[87] 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

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

[89] Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton. ``A Fault-Tolerant Honeycomb Memory''. Quantum 5, 605 (2021).
https://doi.org/10.22331/q-2021-12-20-605

[90] 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

[91] David C. McKay, Ian Hincks, Emily J. Pritchett, Malcolm Carroll, Luke C. G. Govia, and Seth T. Merkel. ``Benchmarking quantum processor performance at scale'' (2023). arXiv:2311.05933.
arXiv:2311.05933

[92] A. Yu. Kitaev. ``Quantum error correction with imperfect gates''. Pages 181–188. Springer US. Boston, MA (1997).
https://doi.org/10.1007/978-1-4615-5923-8_19

[93] David P. DiVincenzo and Peter W. Shor. ``Fault-tolerant error correction with efficient quantum codes''. Phys. Rev. Lett. 77, 3260–3263 (1996).
https://doi.org/10.1103/PhysRevLett.77.3260

[94] A. M. Steane. ``Active stabilization, quantum computation, and quantum state synthesis''. Phys. Rev. Lett. 78, 2252–2255 (1997).
https://doi.org/10.1103/PhysRevLett.78.2252

[95] E. Knill. ``Quantum computing with realistically noisy devices''. Nature 434, 39–44 (2005).
https://doi.org/10.1038/nature03350

[96] Daniel Litinski. ``Magic State Distillation: Not as Costly as You Think''. Quantum 3, 205 (2019).
https://doi.org/10.22331/q-2019-12-02-205

[97] Dominic Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. ``Surface code quantum computing by lattice surgery''. New Journal of Physics 14, 123011 (2012).
https://doi.org/10.1088/1367-2630/14/12/123011

[98] Austin G. Fowler and Craig Gidney. ``Low overhead quantum computation using lattice surgery'' (2019). arXiv:1808.06709.
arXiv:1808.06709

[99] Dave Bacon. ``Operator quantum error-correcting subsystems for self-correcting quantum memories''. Phys. Rev. A 73, 012340 (2006).
https://doi.org/10.1103/PhysRevA.73.012340

[100] H. Bombin. ``Topological subsystem codes''. Phys. Rev. A 81, 032301 (2010).
https://doi.org/10.1103/PhysRevA.81.032301

[101] Muyuan Li, Daniel Miller, Michael Newman, Yukai Wu, and Kenneth R. Brown. ``2d compass codes''. Phys. Rev. X 9, 021041 (2019).
https://doi.org/10.1103/PhysRevX.9.021041

[102] Craig Gidney and Dave Bacon. ``Less bacon more threshold'' (2023). arXiv:2305.12046.
arXiv:2305.12046

[103] Craig Gidney. ``A Pair Measurement Surface Code on Pentagons''. Quantum 7, 1156 (2023).
https://doi.org/10.22331/q-2023-10-25-1156

[104] 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).
https://doi.org/10.22331/q-2024-08-02-1429

[105] D. Ristè, J. G. van Leeuwen, H.-S. Ku, K. W. Lehnert, and L. DiCarlo. ``Initialization by measurement of a superconducting quantum bit circuit''. Phys. Rev. Lett. 109, 050507 (2012).
https://doi.org/10.1103/PhysRevLett.109.050507

[106] D. Ristè, M. Dukalski, C. A. Watson, G. de Lange, M. J. Tiggelman, Ya. M. Blanter, K. W. Lehnert, R. N. Schouten, and L. DiCarlo. ``Deterministic entanglement of superconducting qubits by parity measurement and feedback''. Nature 502, 350–354 (2013).
https://doi.org/10.1038/nature12513

[107] William P. Livingston, Machiel S. Blok, Emmanuel Flurin, Justin Dressel, Andrew N. Jordan, and Irfan Siddiqi. ``Experimental demonstration of continuous quantum error correction''. Nature Communications 13, 2307 (2022).
https://doi.org/10.1038/s41467-022-29906-0

[108] Daniel Gottesman. ``An introduction to quantum error correction and fault-tolerant quantum computation'' (2009). arXiv:0904.2557.
arXiv:0904.2557

[109] 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

[110] Daniel Gottesman. ``The heisenberg representation of quantum computers'' (1998). arXiv:quant-ph/9807006.
arXiv:quant-ph/9807006

[111] Scott Aaronson and Daniel Gottesman. ``Improved simulation of stabilizer circuits''. Phys. Rev. A 70, 052328 (2004).
https://doi.org/10.1103/PhysRevA.70.052328

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

[113] Oscar Higgott and Craig Gidney. ``Sparse Blossom: correcting a million errors per core second with minimum-weight matching''. Quantum 9, 1600 (2025).
https://doi.org/10.22331/q-2025-01-20-1600

[114] Alex Townsend-Teague, Julio Magdalena de la Fuente, and Markus Kesselring. ``Floquetifying the colour code''. Electronic Proceedings in Theoretical Computer Science 384, 265–303 (2023).
https://doi.org/10.4204/eptcs.384.14

[115] Panos Aliferis and John Preskill. ``Fault-tolerant quantum computation against biased noise''. Phys. Rev. A 78, 052331 (2008).
https://doi.org/10.1103/PhysRevA.78.052331

[116] David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia. ``Ultrahigh error threshold for surface codes with biased noise''. Phys. Rev. Lett. 120, 050505 (2018).
https://doi.org/10.1103/PhysRevLett.120.050505

[117] W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel. ``Standard forms of noisy quantum operations via depolarization''. Phys. Rev. A 72, 052326 (2005).
https://doi.org/10.1103/PhysRevA.72.052326

[118] Zhenyu Cai and Simon C. Benjamin. ``Constructing smaller pauli twirling sets for arbitrary error channels''. Scientific Reports 9, 11281 (2019).
https://doi.org/10.1038/s41598-019-46722-7

[119] Michael R. Geller and Zhongyuan Zhou. ``Efficient error models for fault-tolerant architectures and the pauli twirling approximation''. Phys. Rev. A 88, 012314 (2013).
https://doi.org/10.1103/PhysRevA.88.012314

[120] Zijun Chen, Kevin J. Satzinger, Juan Atalaya, Alexander N. Korotkov, Andrew Dunsworth, Daniel Sank, Chris Quintana, Matt McEwen, Rami Barends, Paul V. Klimov, Sabrina Hong, Cody Jones, Andre Petukhov, Dvir Kafri, Sean Demura, Brian Burkett, Craig Gidney, Austin G. Fowler, Alexandru Paler, Harald Putterman, Igor Aleiner, Frank Arute, Kunal Arya, Ryan Babbush, Joseph C. Bardin, Andreas Bengtsson, Alexandre Bourassa, Michael Broughton, Bob B. Buckley, David A. Buell, Nicholas Bushnell, Benjamin Chiaro, Roberto Collins, William Courtney, Alan R. Derk, Daniel Eppens, Catherine Erickson, Edward Farhi, Brooks Foxen, Marissa Giustina, Ami Greene, Jonathan A. Gross, Matthew P. Harrigan, Sean D. Harrington, Jeremy Hilton, Alan Ho, Trent Huang, William J. Huggins, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Kostyantyn Kechedzhi, Seon Kim, Alexei Kitaev, Fedor Kostritsa, David Landhuis, Pavel Laptev, Erik Lucero, Orion Martin, Jarrod R. McClean, Trevor McCourt, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Michael Newman, Murphy Yuezhen Niu, Thomas E. O'Brien, Alex Opremcak, Eric Ostby, Bálint Pató, Nicholas Redd, Pedram Roushan, Nicholas C. Rubin, Vladimir Shvarts, Doug Strain, Marco Szalay, Matthew D. Trevithick, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Adam Zalcman, Hartmut Neven, Sergio Boixo, Vadim Smelyanskiy, Yu Chen, Anthony Megrant, Julian Kelly, and Google Quantum AI. ``Exponential suppression of bit or phase errors with cyclic error correction''. Nature 595, 383–387 (2021).
https://doi.org/10.1038/s41586-021-03588-y

[121] Bibek Pokharel and Daniel A. Lidar. ``Demonstration of algorithmic quantum speedup''. Phys. Rev. Lett. 130, 210602 (2023).
https://doi.org/10.1103/PhysRevLett.130.210602

[122] Hui Khoon Ng, Daniel A. Lidar, and John Preskill. ``Combining dynamical decoupling with fault-tolerant quantum computation''. Phys. Rev. A 84, 012305 (2011).
https://doi.org/10.1103/PhysRevA.84.012305

[123] Gerardo A. Paz-Silva and D. A. Lidar. ``Optimally combining dynamical decoupling and quantum error correction''. Scientific Reports 3, 1530 (2013).
https://doi.org/10.1038/srep01530

[124] James R Wootton. ``Hexagonal matching codes with two-body measurements''. Journal of Physics A: Mathematical and Theoretical 55, 295302 (2022).
https://doi.org/10.1088/1751-8121/ac7a75

[125] James R. Wootton. ``Measurements of floquet code plaquette stabilizers'' (2022). arXiv:2210.13154.
https://doi.org/10.5451/unibas-ep96333
arXiv:2210.13154

Cited by

[1] Jahan Claes, "Dynamic circuit for the honeycomb Floquet code", Physical Review A 112 6, 062406 (2025).

[2] Younghun Kim, Martin Sevior, and Muhammad Usman, "Magic state injection on IBM quantum processors above the distillation threshold", Scientific Reports 16 1, 11189 (2026).

[3] Hansol Kim, Wonjae Choi, and Younghun Kwon, "Implementation of magic state injection within heavy-hexagon architectures", Quantum Information Processing 24 12, 384 (2025).

[4] Younghun Kim, Martin Sevior, and Muhammad Usman, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 482 (2025) ISBN:979-8-3315-5736-2.

[5] Elijah Pelofske, Marek M. Rams, Andreas Bärtschi, Piotr Czarnik, Paolo Braccia, Lukasz Cincio, and Stephan Eidenbenz, "Evaluating the limits of Quantum Approximate Optimization Algorithm parameter transfer at high rounds on sparse Ising models with geometrically local cubic terms", Physical Review Research 8 2, 023023 (2026).

[6] Aziz Kerem Özkan, Muhammet Talha Kakız, Erkan Güler, and Tuğrul Çavdar, 2025 18th International Conference on Information Security and Cryptology (ISCTürkiye) 1 (2025) ISBN:979-8-3315-5709-6.

[7] Aleksandra Świerkowska, Jannik Pflieger, Emmanouil Giortamis, and Pramod Bhatotia, Abstracts of the 2026 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems 201 (2026) ISBN:9798400725708.

[8] Aleksandra Świerkowska, Jannik Pflieger, Emmanouil Giortamis, and Pramod Bhatotia, "ECCentric: An Empirical Analysis of Quantum Error Correction Codes", ACM SIGMETRICS Performance Evaluation Review 54 1, 201 (2026).

[9] Hansol Kim, Wonjae Choi, and Younghun Kwon, "About Implementation of Magic State Injection in Heavy-Hexagon Structure", Mathematics 13 23, 3874 (2025).

[10] Alican Caglar, Imri Fattal, Clement Godfrin, Roy Li, Steven Van Winckel, Kristiaan De Greve, Piet Wambacq, and Jan Craninckx, "A Scalable mK DC Demultiplexer With Extremely Low OFF-Leakage CMOS Switches for Biasing of Spin Qubits", IEEE Solid-State Circuits Letters 8, 221 (2025).

[11] Yuqian Huo, David Quiroga, Anastasios Kyrillidis, and Tirthak Patel, "Three Birds with One Stone: Improving Performance, Convergence, and System Throughput with NEST", Proceedings of the ACM on Measurement and Analysis of Computing Systems 9 3, 1 (2025).

[12] Elijah Pelofske, Andreas Bärtschi, and Stephan Eidenbenz, "Classical combinatorial optimization scaling for random Ising models on 2D heavy-hex graphs", Journal of Physics Communications 10 5, 055003 (2026).

[13] Shao-Hen Chiew, Ezequiel Ignacio Rodríguez Chiacchio, Vishal Sharma, Jing Hao Chai, and Hui Khoon Ng, "Fault-tolerant quantum circuits on connectivity-constrained hardware with swap gates", Physical Review Research 8 3, 033081 (2026).

[14] Younghun Kim, Hansol Kim, Jeongsoo Kang, Wonjae Choi, and Younghun Kwon, "Effectiveness of the syndrome extraction circuit with flag qubits on IBM quantum hardware", Quantum 9, 1893 (2025).

[15] Younghun Kim, Martin Sevior, and Muhammad Usman, "Transversal cnot gate with multicycle error correction", Physical Review Applied 23 2, 024074 (2025).

[16] Bence Hetényi and James R. Wootton, "Creating Entangled Logical Qubits in the Heavy-Hex Lattice with Topological Codes", PRX Quantum 5 4, 040334 (2024).

[17] Younghun Kim, Martin Sevior, and Muhammad Usman, "Magic State Injection on IBM Quantum Processors Above the Distillation Threshold", arXiv:2412.01446, (2024).

[18] Younghun Kim, Hansol Kim, Jeongsoo Kang, Wonjae Choi, and Younghun Kwon, "Effectiveness of the syndrome extraction circuit with flag qubits on IBM quantum hardware", arXiv:2403.10217, (2024).

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

[20] Aleksandra Świerkowska, Jannik Pflieger, Emmanouil Giortamis, and Pramod Bhatotia, "ECCentric: An Empirical Analysis of Quantum Error Correction Codes", arXiv:2511.01062, (2025).

[21] Yuchen Zhu, Jinglei Cheng, Boxi Li, Yidong Zhou, Yufei Ding, and Zhiding Liang, "Leveraging Hardware Power through Optimal Pulse Profiling for Each Qubit Pair", arXiv:2411.19308, (2024).

[22] César Benito, Alfredo Ricci Vasquez, Jonathan Home, Karan K. Mehta, Thomas Monz, Markus Müller, and Alejandro Bermudez, "Scaling roadmap for modular trapped-ion QEC and lattice-surgery teleportation", arXiv:2512.20435, (2025).

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

[24] Dominik Köster and Wolfgang Mauerer, "Claim against Measurement: Statistical Artefacts in Quantum Error Mitigation Benchmarks", arXiv:2605.29872, (2026).

[25] Younghun Kim, Spiro Gicev, Martin Sevior, and Muhammad Usman, "LUCI on IBM Hardware: Error Suppression with Almost Half Syndrome Density", arXiv:2607.01887, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-19 15:47:13) and SAO/NASA ADS (last updated successfully 2026-08-19 15:47:14). 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.