LUCI in the Surface Code with Dropouts
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AbstractRecently, usage of detecting regions facilitated the discovery of new circuits for fault-tolerantly implementing the surface code. Building on these ideas, we present LUCI, a framework for constructing fault-tolerant circuits flexible enough to construct aperiodic and anisotropic circuits, making it a clear step towards quantum error correction beyond static codes. We show that LUCI can be used to adapt surface code circuits to lattices with imperfect qubit and coupler yield, a key challenge for fault-tolerant quantum computers using solid-state architectures. These circuits preserve spacelike distance for isolated broken couplers or isolated broken measure qubits in exchange for halving timelike distance, substantially reducing the penalty for dropout compared to the state of the art and creating opportunities in device architecture design. For qubit and coupler dropout rates of 1% and a patch diameter of 15, LUCI achieves an average spacelike distance of 13.1, compared to 9.1 for the best method in the literature. For a SI1000(0.001) circuit noise model, this translates to a 36x improvement in median logical error rate per round, a factor which increases with device performance. At these dropout and error rates, LUCI requires roughly 25% fewer physical qubits to reach algorithmically relevant one-in-a-trillion logical codeblock error rates.Featured image: LUCI Diagrams and their circuit interpretation. (left) LUCI diagram showing the two rounds of a distance-5 three-coupler surface code circuit [10]. Colors indicate the mid-cycle stabilizers of the surface code being measured, with X stabilizers in red and Z stabilizers in blue. The gray shapes inside the squares indicate how that stabilizer will be measured. (right) Circuit compilations for U-shapes on X-type (red) and Z-type (blue) squares. The colored regions depict the detecting region contracting and being measured out, and then a new detecting region expanding from a reset. Shades of gray in the shape denote which circuit layer the operation occurs in. Other shapes, like the L’s on the top and left boundaries, are formed by removing the appropriate CNOT gates from these U-shape compilations. Popular summaryThe surface code is the leading quantum error correcting code for most solid state architectures. Recent progress has expanded the space of possible circuits we use to implement the surface code in experiment. In this work we introduce a near-infinite family of surface code circuits using a framework called LUCI. We show that these LUCI circuits can exists on lattices of qubits that are subsets of the usual square lattice, allowing us to implement surface code circuits on quantum computers with defective qubits or couplers with better performance than the state of the art.► BibTeX data@article{Debroy2025luciinsurfacecode, doi = {10.22331/q-2025-12-11-1936}, url = {https://doi.org/10.22331/q-2025-12-11-1936}, title = {{LUCI} in the {S}urface {C}ode with {D}ropouts}, author = {Debroy, Dripto M. and McEwen, Matt and Gidney, Craig and Shutty, Noah and Zalcman, Adam}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1936}, month = dec, year = {2025} }► References [1] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer. ``Quantum computing enhanced computational catalysis''. Phys. Rev. Res. 3, 033055 (2021). https://doi.org/10.1103/PhysRevResearch.3.033055 [2] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush. ``Even more efficient quantum computations of chemistry through tensor hypercontraction''. PRX Quantum 2, 030305 (2021). https://doi.org/10.1103/PRXQuantum.2.030305 [3] Isaac H. Kim, Ye-Hua Liu, Sam Pallister, William Pol, Sam Roberts, and Eunseok Lee. ``Fault-tolerant resource estimate for quantum chemical simulations: Case study on li-ion battery electrolyte molecules''. Phys. Rev. Res. 4, 023019 (2022). https://doi.org/10.1103/PhysRevResearch.4.023019 [4] Modjtaba Shokrian Zini, Alain Delgado, Roberto dos Reis, Pablo Antonio Moreno Casares, Jonathan E. Mueller, Arne-Christian Voigt, and Juan Miguel Arrazola. ``Quantum simulation of battery materials using ionic pseudopotentials''. Quantum 7, 1049 (2023). https://doi.org/10.22331/q-2023-07-10-1049 [5] Nicholas C. Rubin, Dominic W. Berry, Fionn D. Malone, Alec F. White, Tanuj Khattar, A. Eugene DePrince, Sabrina Sicolo, Michael Küehn, Michael Kaicher, Joonho Lee, and Ryan Babbush. ``Fault-tolerant quantum simulation of materials using bloch orbitals''. PRX Quantum 4, 040303 (2023). https://doi.org/10.1103/PRXQuantum.4.040303 [6] Google Quantum AI. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5 [7] Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, Ming Gong, Cheng Guo, Chu Guo, Shaojun Guo, Lianchen Han, Linyin Hong, He-Liang Huang, Yong-Heng Huo, Liping Li, Na Li, Shaowei Li, Yuan Li, Futian Liang, Chun Lin, Jin Lin, Haoran Qian, Dan Qiao, Hao Rong, Hong Su, Lihua Sun, Liangyuan Wang, Shiyu Wang, Dachao Wu, Yu Xu, Kai Yan, Weifeng Yang, Yang Yang, Yangsen Ye, Jianghan Yin, Chong Ying, Jiale Yu, Chen Zha, Cha Zhang, Haibin Zhang, Kaili Zhang, Yiming Zhang, Han Zhao, Youwei Zhao, Liang Zhou, Qingling Zhu, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, and Jian-Wei Pan. ``Strong quantum computational advantage using a superconducting quantum processor''. Phys. Rev. Lett. 127, 180501 (2021). https://doi.org/10.1103/PhysRevLett.127.180501 [8] Qingling Zhu, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, Ming Gong, et al. ``Quantum computational advantage via 60-qubit 24-cycle random circuit sampling''. Science bulletin 67, 240–245 (2022). https://doi.org/10.1016/j.scib.2021.10.017 [9] P. V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Yu Chen, A. Dunsworth, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, T. Huang, E. Jeffrey, Erik Lucero, J. Y. Mutus, O. Naaman, C. Neill, C. Quintana, P. Roushan, Daniel Sank, A. Vainsencher, J. Wenner, T. C. White, S. Boixo, R. Babbush, V. N. Smelyanskiy, H. Neven, and John M. Martinis. ``Fluctuations of energy-relaxation times in superconducting qubits''. Phys. Rev. Lett. 121, 090502 (2018). https://doi.org/10.1103/PhysRevLett.121.090502 [10] 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 [11] 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). https://doi.org/10.1103/PhysRevA.96.042316 [12] Armands Strikis, Simon C. Benjamin, and Benjamin J. Brown. ``Quantum computing is scalable on a planar array of qubits with fabrication defects''. Phys. Rev. Appl. 19, 064081 (2023). https://doi.org/10.1103/PhysRevApplied.19.064081 [13] Adam Siegel, Armands Strikis, Thomas Flatters, and Simon Benjamin. ``Adaptive surface code for quantum error correction in the presence of temporary or permanent defects''. Quantum 7, 1065 (2023). https://doi.org/10.22331/q-2023-07-25-1065 [14] Sophia Fuhui Lin, Joshua Viszlai, Kaitlin N. Smith, Gokul Subramanian Ravi, Charles Yuan, Frederic T. Chong, and Benjamin J. Brown. ``Codesign of quantum error-correcting codes and modular chiplets in the presence of defects''. In Proceedings of the 29th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 2. Page 216–231. ASPLOS '24New York, NY, USA (2024). Association for Computing Machinery. https://doi.org/10.1145/3620665.3640362 [15] Zuolin Wei, Tan He, Yangsen Ye, Dachao Wu, Yiming Zhang, Youwei Zhao, Weiping Lin, He-Liang Huang, Xiaobo Zhu, and Jian-Wei Pan. ``Low-overhead defect-adaptive surface code with bandage-like super-stabilizers''. npj Quantum Information 11, 75 (2025). https://doi.org/10.1038/s41534-025-01023-y [16] 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 [17] Joan Camps, Ophelia Crawford, György P. Gehér, Alexander V. Gramolin, Matthew P. Stafford, and Mark Turner. ``Leakage mobility in superconducting qubits as a leakage reduction unit'' (2024). arXiv:2406.04083. arXiv:2406.04083 [18] Mackenzie H. Shaw and Barbara M. Terhal. ``Lowering connectivity requirements for bivariate bicycle codes using morphing circuits'' (2025). https://doi.org/10.1103/PhysRevLett.134.090602 [19] Stefanie J. Beale and Joel J. Wallman. ``Randomized compiling in fault-tolerant quantum computation'' (2023). arXiv:2306.13752. arXiv:2306.13752 [20] Aditya Jain, Pavithran Iyer, Stephen D. Bartlett, and Joseph Emerson. ``Improved quantum error correction with randomized compiling''. Phys. Rev. Res. 5, 033049 (2023). https://doi.org/10.1103/PhysRevResearch.5.033049 [21] 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 [22] Craig Gidney. ``Stability Experiments: The Overlooked Dual of Memory Experiments''. Quantum 6, 786 (2022). https://doi.org/10.22331/q-2022-08-24-786 [23] Dripto M. Debroy. ``Resources for "luci in the surface code with defects"'' (2024). [24] Oscar Higgott and Nikolas P. Breuckmann. ``Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead''. Phys. Rev. X 11, 031039 (2021). https://doi.org/10.1103/PhysRevX.11.031039 [25] Noah Shutty, Michael Newman, and Benjamin Villalonga. ``Efficient near-optimal decoding of the surface code through ensembling'' (2024). arXiv:2401.12434. arXiv:2401.12434 [26] Cody Jones. ``Improved accuracy for decoding surface codes with matching synthesis'' (2024). arXiv:2408.12135. arXiv:2408.12135 [27] Naomi Nickerson and Héctor Bombín. ``Measurement based fault tolerance beyond foliation'' (2018). arXiv:1810.09621. arXiv:1810.09621 [28] Michael Newman, Leonardo Andreta de Castro, and Kenneth R. Brown. ``Generating Fault-Tolerant Cluster States from Crystal Structures''. Quantum 4, 295 (2020). https://doi.org/10.22331/q-2020-07-13-295 [29] Matthew B. Hastings and Jeongwan Haah. ``Dynamically Generated Logical Qubits''. Quantum 5, 564 (2021). https://doi.org/10.22331/q-2021-10-19-564 [30] Xiaozhen Fu and Daniel Gottesman. ``Error correction in dynamical codes'' (2024). arXiv:2403.04163. https://doi.org/10.22331/q-2025-10-20-1886 arXiv:2403.04163 [31] Craig Gidney, Michael Newman, and Matt McEwen. ``Benchmarking the Planar Honeycomb Code''. Quantum 6, 813 (2022). https://doi.org/10.22331/q-2022-09-21-813Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-11 11:22:19: Could not fetch cited-by data for 10.22331/q-2025-12-11-1936 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-11 11:22:19: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractRecently, usage of detecting regions facilitated the discovery of new circuits for fault-tolerantly implementing the surface code. Building on these ideas, we present LUCI, a framework for constructing fault-tolerant circuits flexible enough to construct aperiodic and anisotropic circuits, making it a clear step towards quantum error correction beyond static codes. We show that LUCI can be used to adapt surface code circuits to lattices with imperfect qubit and coupler yield, a key challenge for fault-tolerant quantum computers using solid-state architectures. These circuits preserve spacelike distance for isolated broken couplers or isolated broken measure qubits in exchange for halving timelike distance, substantially reducing the penalty for dropout compared to the state of the art and creating opportunities in device architecture design. For qubit and coupler dropout rates of 1% and a patch diameter of 15, LUCI achieves an average spacelike distance of 13.1, compared to 9.1 for the best method in the literature. For a SI1000(0.001) circuit noise model, this translates to a 36x improvement in median logical error rate per round, a factor which increases with device performance. At these dropout and error rates, LUCI requires roughly 25% fewer physical qubits to reach algorithmically relevant one-in-a-trillion logical codeblock error rates.Featured image: LUCI Diagrams and their circuit interpretation. (left) LUCI diagram showing the two rounds of a distance-5 three-coupler surface code circuit [10]. Colors indicate the mid-cycle stabilizers of the surface code being measured, with X stabilizers in red and Z stabilizers in blue. The gray shapes inside the squares indicate how that stabilizer will be measured. (right) Circuit compilations for U-shapes on X-type (red) and Z-type (blue) squares. The colored regions depict the detecting region contracting and being measured out, and then a new detecting region expanding from a reset. Shades of gray in the shape denote which circuit layer the operation occurs in. Other shapes, like the L’s on the top and left boundaries, are formed by removing the appropriate CNOT gates from these U-shape compilations. Popular summaryThe surface code is the leading quantum error correcting code for most solid state architectures. Recent progress has expanded the space of possible circuits we use to implement the surface code in experiment. In this work we introduce a near-infinite family of surface code circuits using a framework called LUCI. We show that these LUCI circuits can exists on lattices of qubits that are subsets of the usual square lattice, allowing us to implement surface code circuits on quantum computers with defective qubits or couplers with better performance than the state of the art.► BibTeX data@article{Debroy2025luciinsurfacecode, doi = {10.22331/q-2025-12-11-1936}, url = {https://doi.org/10.22331/q-2025-12-11-1936}, title = {{LUCI} in the {S}urface {C}ode with {D}ropouts}, author = {Debroy, Dripto M. and McEwen, Matt and Gidney, Craig and Shutty, Noah and Zalcman, Adam}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1936}, month = dec, year = {2025} }► References [1] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer. ``Quantum computing enhanced computational catalysis''. Phys. Rev. Res. 3, 033055 (2021). https://doi.org/10.1103/PhysRevResearch.3.033055 [2] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush. ``Even more efficient quantum computations of chemistry through tensor hypercontraction''. PRX Quantum 2, 030305 (2021). https://doi.org/10.1103/PRXQuantum.2.030305 [3] Isaac H. Kim, Ye-Hua Liu, Sam Pallister, William Pol, Sam Roberts, and Eunseok Lee. ``Fault-tolerant resource estimate for quantum chemical simulations: Case study on li-ion battery electrolyte molecules''. Phys. Rev. Res. 4, 023019 (2022). https://doi.org/10.1103/PhysRevResearch.4.023019 [4] Modjtaba Shokrian Zini, Alain Delgado, Roberto dos Reis, Pablo Antonio Moreno Casares, Jonathan E. Mueller, Arne-Christian Voigt, and Juan Miguel Arrazola. ``Quantum simulation of battery materials using ionic pseudopotentials''. Quantum 7, 1049 (2023). https://doi.org/10.22331/q-2023-07-10-1049 [5] Nicholas C. Rubin, Dominic W. Berry, Fionn D. Malone, Alec F. White, Tanuj Khattar, A. Eugene DePrince, Sabrina Sicolo, Michael Küehn, Michael Kaicher, Joonho Lee, and Ryan Babbush. ``Fault-tolerant quantum simulation of materials using bloch orbitals''. PRX Quantum 4, 040303 (2023). https://doi.org/10.1103/PRXQuantum.4.040303 [6] Google Quantum AI. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5 [7] Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, Ming Gong, Cheng Guo, Chu Guo, Shaojun Guo, Lianchen Han, Linyin Hong, He-Liang Huang, Yong-Heng Huo, Liping Li, Na Li, Shaowei Li, Yuan Li, Futian Liang, Chun Lin, Jin Lin, Haoran Qian, Dan Qiao, Hao Rong, Hong Su, Lihua Sun, Liangyuan Wang, Shiyu Wang, Dachao Wu, Yu Xu, Kai Yan, Weifeng Yang, Yang Yang, Yangsen Ye, Jianghan Yin, Chong Ying, Jiale Yu, Chen Zha, Cha Zhang, Haibin Zhang, Kaili Zhang, Yiming Zhang, Han Zhao, Youwei Zhao, Liang Zhou, Qingling Zhu, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, and Jian-Wei Pan. ``Strong quantum computational advantage using a superconducting quantum processor''. Phys. Rev. Lett. 127, 180501 (2021). https://doi.org/10.1103/PhysRevLett.127.180501 [8] Qingling Zhu, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, Ming Gong, et al. ``Quantum computational advantage via 60-qubit 24-cycle random circuit sampling''. Science bulletin 67, 240–245 (2022). https://doi.org/10.1016/j.scib.2021.10.017 [9] P. V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Yu Chen, A. Dunsworth, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, T. Huang, E. Jeffrey, Erik Lucero, J. Y. Mutus, O. Naaman, C. Neill, C. Quintana, P. Roushan, Daniel Sank, A. Vainsencher, J. Wenner, T. C. White, S. Boixo, R. Babbush, V. N. Smelyanskiy, H. Neven, and John M. Martinis. ``Fluctuations of energy-relaxation times in superconducting qubits''. Phys. Rev. Lett. 121, 090502 (2018). https://doi.org/10.1103/PhysRevLett.121.090502 [10] 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 [11] 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). https://doi.org/10.1103/PhysRevA.96.042316 [12] Armands Strikis, Simon C. Benjamin, and Benjamin J. Brown. ``Quantum computing is scalable on a planar array of qubits with fabrication defects''. Phys. Rev. Appl. 19, 064081 (2023). https://doi.org/10.1103/PhysRevApplied.19.064081 [13] Adam Siegel, Armands Strikis, Thomas Flatters, and Simon Benjamin. ``Adaptive surface code for quantum error correction in the presence of temporary or permanent defects''. Quantum 7, 1065 (2023). https://doi.org/10.22331/q-2023-07-25-1065 [14] Sophia Fuhui Lin, Joshua Viszlai, Kaitlin N. Smith, Gokul Subramanian Ravi, Charles Yuan, Frederic T. Chong, and Benjamin J. Brown. ``Codesign of quantum error-correcting codes and modular chiplets in the presence of defects''. In Proceedings of the 29th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 2. Page 216–231. ASPLOS '24New York, NY, USA (2024). Association for Computing Machinery. https://doi.org/10.1145/3620665.3640362 [15] Zuolin Wei, Tan He, Yangsen Ye, Dachao Wu, Yiming Zhang, Youwei Zhao, Weiping Lin, He-Liang Huang, Xiaobo Zhu, and Jian-Wei Pan. ``Low-overhead defect-adaptive surface code with bandage-like super-stabilizers''. npj Quantum Information 11, 75 (2025). https://doi.org/10.1038/s41534-025-01023-y [16] 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 [17] Joan Camps, Ophelia Crawford, György P. Gehér, Alexander V. Gramolin, Matthew P. Stafford, and Mark Turner. ``Leakage mobility in superconducting qubits as a leakage reduction unit'' (2024). arXiv:2406.04083. arXiv:2406.04083 [18] Mackenzie H. Shaw and Barbara M. Terhal. ``Lowering connectivity requirements for bivariate bicycle codes using morphing circuits'' (2025). https://doi.org/10.1103/PhysRevLett.134.090602 [19] Stefanie J. Beale and Joel J. Wallman. ``Randomized compiling in fault-tolerant quantum computation'' (2023). arXiv:2306.13752. arXiv:2306.13752 [20] Aditya Jain, Pavithran Iyer, Stephen D. Bartlett, and Joseph Emerson. ``Improved quantum error correction with randomized compiling''. Phys. Rev. Res. 5, 033049 (2023). https://doi.org/10.1103/PhysRevResearch.5.033049 [21] 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 [22] Craig Gidney. ``Stability Experiments: The Overlooked Dual of Memory Experiments''. Quantum 6, 786 (2022). https://doi.org/10.22331/q-2022-08-24-786 [23] Dripto M. Debroy. ``Resources for "luci in the surface code with defects"'' (2024). [24] Oscar Higgott and Nikolas P. Breuckmann. ``Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead''. Phys. Rev. X 11, 031039 (2021). https://doi.org/10.1103/PhysRevX.11.031039 [25] Noah Shutty, Michael Newman, and Benjamin Villalonga. ``Efficient near-optimal decoding of the surface code through ensembling'' (2024). arXiv:2401.12434. arXiv:2401.12434 [26] Cody Jones. ``Improved accuracy for decoding surface codes with matching synthesis'' (2024). arXiv:2408.12135. arXiv:2408.12135 [27] Naomi Nickerson and Héctor Bombín. ``Measurement based fault tolerance beyond foliation'' (2018). arXiv:1810.09621. arXiv:1810.09621 [28] Michael Newman, Leonardo Andreta de Castro, and Kenneth R. Brown. ``Generating Fault-Tolerant Cluster States from Crystal Structures''. Quantum 4, 295 (2020). https://doi.org/10.22331/q-2020-07-13-295 [29] Matthew B. Hastings and Jeongwan Haah. ``Dynamically Generated Logical Qubits''. Quantum 5, 564 (2021). https://doi.org/10.22331/q-2021-10-19-564 [30] Xiaozhen Fu and Daniel Gottesman. ``Error correction in dynamical codes'' (2024). arXiv:2403.04163. https://doi.org/10.22331/q-2025-10-20-1886 arXiv:2403.04163 [31] Craig Gidney, Michael Newman, and Matt McEwen. ``Benchmarking the Planar Honeycomb Code''. Quantum 6, 813 (2022). https://doi.org/10.22331/q-2022-09-21-813Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-11 11:22:19: Could not fetch cited-by data for 10.22331/q-2025-12-11-1936 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-11 11:22:19: No response from ADS or unable to decode the received json data when getting the list of citing works.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.