Efficient and high-performance routing of lattice-surgery paths on three-dimensional lattice
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AbstractEncoding logical qubits with surface codes and performing multi-qubit logical operations with lattice surgery is one of the most promising approaches to demonstrate fault-tolerant quantum computing. Thus, a method to efficiently schedule a sequence of lattice-surgery operations is vital for high-performance fault-tolerant quantum computing. A possible strategy to improve the throughput of lattice-surgery operations is splitting a large instruction into several small instructions, such as Bell state preparation and measurements, and executing a part of them in advance. However, scheduling methods to fully utilize this idea have yet to be explored. In this paper, we propose a fast and high-performance scheduling algorithm for lattice-surgery instructions leveraging this strategy. We achieved this by converting the scheduling problem of lattice-surgery instructions to a graph problem of embedding 3D paths into a 3D lattice, which enables us to explore efficient scheduling by solving path search problems in the 3D lattice. Based on this reduction, we propose a method to solve the path-finding problems, the look-ahead Dijkstra projection. We numerically show that this method reduced the execution time of benchmark programs generated from quantum phase estimation algorithms by 3.8 times compared with a naive method based on greedy algorithms. Our study establishes the relation between the lattice-surgery scheduling and graph search problems, which leads to further theoretical analysis on compiler optimization of fault-tolerant quantum computing.Featured image: Overview of the optimization by routing lattice-surgery paths more flexibly in a 3D lattice. The equality of quantum circuits holds up to appropriate feedback of Pauli operations.Popular summarySince qubits are prone to errors, using them directly for large-scale practical problems is not a realistic approach. To fundamentally address the high error rate, we can employ fault-tolerant quantum computing (FTQC), which corrects errors during computation. One promising approach to FTQC implementation is the surface code, which maps qubits on cells of a 2D lattice. In the surface-code architecture, operations on multiple qubits can be achieved by lattice surgery, which temporarily connects target cells through vacant cells. Since the lattice-surgery operations can be parallelized under topological constraints, an algorithm to efficiently schedule the operations is a vital issue to accelerate the execution of FTQC programs. While decomposing operations into smaller steps for parallelization is a common approach to speed up classical computing, its quantum counterpart has not been sufficiently explored. In this work, we consider flexibly splitting lattice-surgery operations into smaller fragments for further parallelization. One of our key findings is that the complicated problem of splitting and scheduling lattice-surgery operations can be reduced to a simple graph problem of embedding 3D paths into a 3D lattice. Utilizing the reduction to the graph problem, we propose an efficient and high-performance lattice-surgery scheduling algorithm, which shows 3.8 times speed-up with real applications compared to the baseline solutions. Our work substantially improves FTQC execution times by incorporating critical techniques in classical computing into the FTQC optimization and reducing the scheduling problem to a simple graph problem. It also opens up the possibility of further FTQC acceleration under deep collaboration between graph algorithms and computer science.► BibTeX data@article{Hamada2026efficienthigh, doi = {10.22331/q-2026-04-13-2061}, url = {https://doi.org/10.22331/q-2026-04-13-2061}, title = {Efficient and high-performance routing of lattice-surgery paths on three-dimensional lattice}, author = {Hamada, Kou and Suzuki, Yasunari and Tokunaga, Yuuki}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2061}, month = apr, year = {2026} }► References [1] Peter W. Shor. ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer''. SIAM Review 41, 303–332 (1999). doi: 10.1137/S0036144598347011. https://doi.org/10.1137/S0036144598347011 [2] A. Yu. Kitaev. ``Quantum measurements and the abelian stabilizer problem'' (1995). doi: 10.48550/arXiv.quant-ph/9511026. arXiv:quant-ph/9511026. https://doi.org/10.48550/arXiv.quant-ph/9511026 arXiv:quant-ph/9511026 [3] Seth Lloyd. ``Universal quantum simulators''. Science 273, 1073–1078 (1996). doi: 10.1126/science.273.5278.1073. https://doi.org/10.1126/science.273.5278.1073 [4] A. Yu. Kitaev. ``Quantum computations: algorithms and error correction''.
Russian Mathematical Surveys 52, 1191–1249 (1997). doi: 10.1070/RM1997v052n06ABEH002155. https://doi.org/10.1070/RM1997v052n06ABEH002155 [5] Sergey B. Bravyi and Alexei Yu. Kitaev. ``Quantum codes on a lattice with boundary'' (1998). doi: 10.48550/arXiv.quant-ph/9811052. arXiv:quant-ph/9811052. https://doi.org/10.48550/arXiv.quant-ph/9811052 arXiv:quant-ph/9811052 [6] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Physical Review A 86, 032324 (2012). doi: 10.1103/PhysRevA.86.032324. https://doi.org/10.1103/PhysRevA.86.032324 [7] Clare Horsman, Austin G. Fowler, Simon Devitt, and Rodney Van Meter. ``Surface code quantum computing by lattice surgery''. New Journal of Physics 14, 123011 (2012). doi: 10.1088/1367-2630/14/12/123011. https://doi.org/10.1088/1367-2630/14/12/123011 [8] Austin G. Fowler and Craig Gidney. ``Low overhead quantum computation using lattice surgery'' (2018). doi: doi.org/10.48550/arXiv.1808.06709. arXiv:1808.06709. https://doi.org/10.48550/arXiv.1808.06709 arXiv:1808.06709 [9] Michael E. Beverland, Prakash Murali, Matthias Troyer, Krysta M. Svore, Torsten Hoeffler, Vadym Kliuchnikov, Guang Hao Low, Mathias Soeken, Aarthi Sundaram, and Alexander Vaschillo. ``Assessing requirements to scale to practical quantum advantage'' (2022). doi: 10.48550/arXiv.2211.07629. arXiv:2211.07629. https://doi.org/10.48550/arXiv.2211.07629 arXiv:2211.07629 [10] Ryan Babbush, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven. ``Encoding electronic spectra in quantum circuits with linear t complexity''. Physical Review X 8, 041015 (2018). doi: 10.1103/PhysRevX.8.041015. https://doi.org/10.1103/PhysRevX.8.041015 [11] Nobuyuki Yoshioka, Tsuyoshi Okubo, Yasunari Suzuki, Yuki Koizumi, and Wataru Mizukami. ``Hunting for quantum-classical crossover in condensed matter problems''. npj Quantum Information 10, 45 (2024). doi: 10.1038/s41534-024-00839-4. https://doi.org/10.1038/s41534-024-00839-4 [12] Ali JavadiAbhari, Shruti Patil, Daniel Kudrow, Jeff Heckey, Alexey Lvov, Frederic T. Chong, and Margaret Martonosi. ``ScaffCC: A framework for compilation and analysis of quantum computing programs''. In Proceedings of the 11th ACM Conference on Computing Frontiers. CF '14. Association for Computing Machinery (2014). doi: 10.1145/2597917.2597939. https://doi.org/10.1145/2597917.2597939 [13] Qiskit contributors. ``Qiskit: An open-source framework for quantum computing''. url: doi.org/10.5281/zenodo.2573505. https://doi.org/10.5281/zenodo.2573505 [14] Seyon Sivarajah, Silas Dilkes, Alexander Cowtan, Will Simmons, Alec Edgington, and Ross Duncan. ``t|ket>: a retargetable compiler for NISQ devices''. Quantum Science and Technology 6, 014003 (2020). doi: 10.1088/2058-9565/ab8e92. https://doi.org/10.1088/2058-9565/ab8e92 [15] George Watkins, Hoang Minh Nguyen, Keelan Watkins, Steven Pearce, Hoi-Kwan Lau, and Alexandru Paler. ``A high performance compiler for very large scale surface code computations''. Quantum 8, 1354 (2024). doi: 10.22331/q-2024-05-22-1354. https://doi.org/10.22331/q-2024-05-22-1354 [16] Daniel Herr, Franco Nori, and Simon J. Devitt. ``Optimization of lattice surgery is np-hard''. Npj quantum information 3, 35 (2017). doi: 10.1038/s41534-017-0035-1. https://doi.org/10.1038/s41534-017-0035-1 [17] Abtin Molavi, Amanda Xu, Swamit Tannu, and Aws Albarghouthi. ``Compilation for surface code quantum computers'' (2023). doi: 10.48550/arXiv.2311.18042. arXiv:2311.18042. https://doi.org/10.48550/arXiv.2311.18042 arXiv:2311.18042 [18] Daniel Litinski. ``A game of surface codes: Large-scale quantum computing with lattice surgery''. Quantum 3, 128 (2019). doi: 10.22331/q-2019-03-05-128. https://doi.org/10.22331/q-2019-03-05-128 [19] Daniel Litinski. ``Magic state distillation: Not as costly as you think''. Quantum 3, 205 (2019). doi: 10.22331/q-2019-12-02-205. https://doi.org/10.22331/q-2019-12-02-205 [20] Lingling Lao, Bas van Wee, Imran Ashraf, J. van Someren, Nader Khammassi, Koen Bertels, and Carmen G. Almudever. ``Mapping of lattice surgery-based quantum circuits on surface code architectures''. Quantum Science and Technology 4, 015005 (2018). doi: 10.1088/2058-9565/aadd1a. https://doi.org/10.1088/2058-9565/aadd1a [21] Michael Beverland, Vadym Kliuchnikov, and Eddie Schoute. ``Surface code compilation via edge-disjoint paths''. PRX Quantum 3, 020342 (2022). doi: 10.1103/PRXQuantum.3.020342. https://doi.org/10.1103/PRXQuantum.3.020342 [22] Guang Hao Low and Isaac L. Chuang. ``Hamiltonian simulation by qubitization''. Quantum 3, 163 (2019). doi: 10.22331/q-2019-07-12-163. https://doi.org/10.22331/q-2019-07-12-163 [23] Austin G. Fowler, Adam C. Whiteside, and Lloyd C. L. Hollenberg. ``Towards practical classical processing for the surface code''. Physical review letters 108, 180501 (2012). doi: 10.1103/PhysRevLett.108.180501. https://doi.org/10.1103/PhysRevLett.108.180501 [24] Rajeev Acharya, Igor Aleiner, Richard Allen, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, et al. ``Suppressing quantum errors by scaling a surface code logical qubit''. Nature 614, 676–681 (2023). doi: 10.1038/s41586-022-05434-1. https://doi.org/10.1038/s41586-022-05434-1 [25] Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton. ``Poking holes and cutting corners to achieve clifford gates with the surface code''. Physical Review X 7, 021029 (2017). doi: 10.1103/PhysRevX.7.021029. https://doi.org/10.1103/PhysRevX.7.021029 [26] Michael L. Fredman and Robert Endre Tarjan. ``Fibonacci heaps and their uses in improved network optimization algorithms''. J. ACM 34, 596–615 (1987). doi: 10.1145/28869.28874. https://doi.org/10.1145/28869.28874 [27] Allyson Silva, Xiangyi Zhang, Zak Webb, Mia Kramer, Chan-Woo Yang, Xiao Liu, Jessica Lemieux, Ka-Wai Chen, Artur Scherer, and Pooya Ronagh. ``Multi-qubit lattice surgery scheduling''. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Pages 1–1. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2024). url: doi.org/10.4230/lipics.tqc.2024.1. https://doi.org/10.4230/lipics.tqc.2024.1 [28] Austin G. Fowler. ``Time-optimal quantum computation'' (2012). doi: 10.48550/arXiv.1210.4626. arXiv:1210.4626. https://doi.org/10.48550/arXiv.1210.4626 arXiv:1210.4626 [29] Mingzheng Zhu, Hao Fu, Jun Wu, Chi Zhang, Wei Xie, and Xiang-Yang Li. ``Ecmas: Efficient circuit mapping and scheduling for surface code''. In 2024 IEEE/ACM International Symposium on Code Generation and Optimization (CGO). Page 158–169. IEEE (2024). doi: 10.1109/cgo57630.2024.10444874. https://doi.org/10.1109/cgo57630.2024.10444874 [30] Mingzheng Zhu, Hao Fu, Haishan Song, Jun Wu, Chi Zhang, Wei Xie, and Xiangyang Li. ``Ecmas+: Efficient circuit mapping and scheduling for surface code encoded circuit on quantum cloud platform''. ACM Transactions on Architecture and Code Optimization 22, 1–25 (2025). doi: 10.1145/3760783. https://doi.org/10.1145/3760783 [31] Google Quantum AI. ``Suppressing quantum errors by scaling a surface code logical qubit''. Nature 614, 676–681 (2023). doi: 10.1038/s41586-022-05434-1. https://doi.org/10.1038/s41586-022-05434-1 [32] Google Quantum AI and Collaborators. ``Quantum error correction below the surface code threshold''. Nature 638, 920–926 (2024). doi: 10.1038/s41586-024-08449-y. https://doi.org/10.1038/s41586-024-08449-y [33] Hengyun Zhou, Chen Zhao, Madelyn Cain, Dolev Bluvstein, Nishad Maskara, Casey Duckering, Hong-Ye Hu, Sheng-Tao Wang, Aleksander Kubica, and Mikhail D. Lukin. ``Low-overhead transversal fault tolerance for universal quantum computation''. Nature 646, 303–308 (2025). doi: 10.1038/s41586-025-09543-5. https://doi.org/10.1038/s41586-025-09543-5 [34] Craig Gidney, Michael Newman, Peter Brooks, and Cody Jones. ``Yoked surface codes''. Nature Communications 16 (2025). doi: 10.1038/s41467-025-59714-1. https://doi.org/10.1038/s41467-025-59714-1 [35] Theodore J. Yoder, Eddie Schoute, Patrick Rall, Emily Pritchett, Jay M. Gambetta, Andrew W. Cross, Malcolm Carroll, and Michael E. Beverland. ``Tour de gross: A modular quantum computer based on bivariate bicycle codes'' (2025). doi: 10.48550/ARXIV.2506.03094. https://doi.org/10.48550/ARXIV.2506.03094 [36] Qian Xu, J.
Pablo Bonilla Ataides, Christopher A. Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasić, Mikhail D. Lukin, Liang Jiang, and Hengyun Zhou. ``Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays''. Nature Physics 20, 1084–1090 (2024). doi: 10.1038/s41567-024-02479-z. https://doi.org/10.1038/s41567-024-02479-z [37] Lawrence Z. Cohen, Isaac H. Kim, Stephen D. Bartlett, and Benjamin J. Brown. ``Low-overhead fault-tolerant quantum computing using long-range connectivity''. Science Advances 8 (2022). doi: 10.1126/sciadv.abn1717. https://doi.org/10.1126/sciadv.abn1717 [38] Naomi H Nickerson, Ying Li, and Simon C Benjamin. ``Topological quantum computing with a very noisy network and local error rates approaching one percent''. Nature communications 4, 1756 (2013). url: doi.org/10.1038/ncomms2773. https://doi.org/10.1038/ncomms2773 [39] Hector Bombin, Isaac H Kim, Daniel Litinski, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Sam Roberts, and Terry Rudolph. ``Interleaving: Modular architectures for fault-tolerant photonic quantum computing'' (2021). url: doi.org/10.48550/arXiv.2103.08612. https://doi.org/10.48550/arXiv.2103.08612 [40] J Eli Bourassa, Rafael N Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q Baragiola, Saikat Guha, Guillaume Dauphinais, et al. ``Blueprint for a scalable photonic fault-tolerant quantum computer''. Quantum 5, 392 (2021). url: doi.org/10.22331/q-2021-02-04-392. https://doi.org/10.22331/q-2021-02-04-392 [41] Christopher Pattison, Gefen Baranes, Juan Pablo Bonilla Ataides, Mikhail D Lukin, and Hengyun Zhou. ``Constant-rate entanglement distillation for fast quantum interconnects''. In Proceedings of the 52nd Annual International Symposium on Computer Architecture. Pages 257–270. (2025). url: doi.org/10.1145/3695053.373106. https://doi.org/10.1145/3695053.373106 [42] Yuya Maeda, Yasunari Suzuki, Toshiki Kobayashi, Takashi Yamamoto, Yuuki Tokunaga, and Keisuke Fujii. ``Logical entanglement distribution between distant 2d array qubits'' (2025). url: doi.org/10.48550/arXiv.2503.14894. https://doi.org/10.48550/arXiv.2503.14894 [43] Christopher Chamberland and Earl T Campbell. ``Universal quantum computing with twist-free and temporally encoded lattice surgery''. PRX Quantum 3, 010331 (2022). url: doi.org/10.1103/PRXQuantum.3.010331. https://doi.org/10.1103/PRXQuantum.3.010331 [44] Matt McEwen, Dave Bacon, and Craig Gidney. ``Relaxing hardware requirements for surface code circuits using time-dynamics''. Quantum 7, 1172 (2023). url: doi.org/10.22331/q-2023-11-07-1172. https://doi.org/10.22331/q-2023-11-07-1172 [45] Michael A Nielsen and Isaac L Chuang. ``Quantum computation and quantum information''. Cambridge university press. (2010). url: doi.org/10.1017/CBO9780511976667. https://doi.org/10.1017/CBO9780511976667 [46] Neil J Ross and Peter Selinger. ``Optimal ancilla-free clifford+ t approximation of z-rotations'' (2014). url: doi.org/10.48550/arXiv.1403.2975. https://doi.org/10.48550/arXiv.1403.2975 [47] David P DiVincenzo. ``Quantum gates and circuits''. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 261–276 (1998). url: doi.org/10.1098/rspa.1998.0159. https://doi.org/10.1098/rspa.1998.0159Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-13 17:48:12: Could not fetch cited-by data for 10.22331/q-2026-04-13-2061 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-13 17:48:13: 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. AbstractEncoding logical qubits with surface codes and performing multi-qubit logical operations with lattice surgery is one of the most promising approaches to demonstrate fault-tolerant quantum computing. Thus, a method to efficiently schedule a sequence of lattice-surgery operations is vital for high-performance fault-tolerant quantum computing. A possible strategy to improve the throughput of lattice-surgery operations is splitting a large instruction into several small instructions, such as Bell state preparation and measurements, and executing a part of them in advance. However, scheduling methods to fully utilize this idea have yet to be explored. In this paper, we propose a fast and high-performance scheduling algorithm for lattice-surgery instructions leveraging this strategy. We achieved this by converting the scheduling problem of lattice-surgery instructions to a graph problem of embedding 3D paths into a 3D lattice, which enables us to explore efficient scheduling by solving path search problems in the 3D lattice. Based on this reduction, we propose a method to solve the path-finding problems, the look-ahead Dijkstra projection. We numerically show that this method reduced the execution time of benchmark programs generated from quantum phase estimation algorithms by 3.8 times compared with a naive method based on greedy algorithms. Our study establishes the relation between the lattice-surgery scheduling and graph search problems, which leads to further theoretical analysis on compiler optimization of fault-tolerant quantum computing.Featured image: Overview of the optimization by routing lattice-surgery paths more flexibly in a 3D lattice. The equality of quantum circuits holds up to appropriate feedback of Pauli operations.Popular summarySince qubits are prone to errors, using them directly for large-scale practical problems is not a realistic approach. To fundamentally address the high error rate, we can employ fault-tolerant quantum computing (FTQC), which corrects errors during computation. One promising approach to FTQC implementation is the surface code, which maps qubits on cells of a 2D lattice. In the surface-code architecture, operations on multiple qubits can be achieved by lattice surgery, which temporarily connects target cells through vacant cells. Since the lattice-surgery operations can be parallelized under topological constraints, an algorithm to efficiently schedule the operations is a vital issue to accelerate the execution of FTQC programs. While decomposing operations into smaller steps for parallelization is a common approach to speed up classical computing, its quantum counterpart has not been sufficiently explored. In this work, we consider flexibly splitting lattice-surgery operations into smaller fragments for further parallelization. One of our key findings is that the complicated problem of splitting and scheduling lattice-surgery operations can be reduced to a simple graph problem of embedding 3D paths into a 3D lattice. Utilizing the reduction to the graph problem, we propose an efficient and high-performance lattice-surgery scheduling algorithm, which shows 3.8 times speed-up with real applications compared to the baseline solutions. Our work substantially improves FTQC execution times by incorporating critical techniques in classical computing into the FTQC optimization and reducing the scheduling problem to a simple graph problem. It also opens up the possibility of further FTQC acceleration under deep collaboration between graph algorithms and computer science.► BibTeX data@article{Hamada2026efficienthigh, doi = {10.22331/q-2026-04-13-2061}, url = {https://doi.org/10.22331/q-2026-04-13-2061}, title = {Efficient and high-performance routing of lattice-surgery paths on three-dimensional lattice}, author = {Hamada, Kou and Suzuki, Yasunari and Tokunaga, Yuuki}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2061}, month = apr, year = {2026} }► References [1] Peter W. Shor. ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer''. SIAM Review 41, 303–332 (1999). doi: 10.1137/S0036144598347011. https://doi.org/10.1137/S0036144598347011 [2] A. Yu. Kitaev. ``Quantum measurements and the abelian stabilizer problem'' (1995). doi: 10.48550/arXiv.quant-ph/9511026. arXiv:quant-ph/9511026. https://doi.org/10.48550/arXiv.quant-ph/9511026 arXiv:quant-ph/9511026 [3] Seth Lloyd. ``Universal quantum simulators''. Science 273, 1073–1078 (1996). doi: 10.1126/science.273.5278.1073. https://doi.org/10.1126/science.273.5278.1073 [4] A. Yu. Kitaev. ``Quantum computations: algorithms and error correction''.
Russian Mathematical Surveys 52, 1191–1249 (1997). doi: 10.1070/RM1997v052n06ABEH002155. https://doi.org/10.1070/RM1997v052n06ABEH002155 [5] Sergey B. Bravyi and Alexei Yu. Kitaev. ``Quantum codes on a lattice with boundary'' (1998). doi: 10.48550/arXiv.quant-ph/9811052. arXiv:quant-ph/9811052. https://doi.org/10.48550/arXiv.quant-ph/9811052 arXiv:quant-ph/9811052 [6] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Physical Review A 86, 032324 (2012). doi: 10.1103/PhysRevA.86.032324. https://doi.org/10.1103/PhysRevA.86.032324 [7] Clare Horsman, Austin G. Fowler, Simon Devitt, and Rodney Van Meter. ``Surface code quantum computing by lattice surgery''. New Journal of Physics 14, 123011 (2012). doi: 10.1088/1367-2630/14/12/123011. https://doi.org/10.1088/1367-2630/14/12/123011 [8] Austin G. Fowler and Craig Gidney. ``Low overhead quantum computation using lattice surgery'' (2018). doi: doi.org/10.48550/arXiv.1808.06709. arXiv:1808.06709. https://doi.org/10.48550/arXiv.1808.06709 arXiv:1808.06709 [9] Michael E. Beverland, Prakash Murali, Matthias Troyer, Krysta M. Svore, Torsten Hoeffler, Vadym Kliuchnikov, Guang Hao Low, Mathias Soeken, Aarthi Sundaram, and Alexander Vaschillo. ``Assessing requirements to scale to practical quantum advantage'' (2022). doi: 10.48550/arXiv.2211.07629. arXiv:2211.07629. https://doi.org/10.48550/arXiv.2211.07629 arXiv:2211.07629 [10] Ryan Babbush, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven. ``Encoding electronic spectra in quantum circuits with linear t complexity''. Physical Review X 8, 041015 (2018). doi: 10.1103/PhysRevX.8.041015. https://doi.org/10.1103/PhysRevX.8.041015 [11] Nobuyuki Yoshioka, Tsuyoshi Okubo, Yasunari Suzuki, Yuki Koizumi, and Wataru Mizukami. ``Hunting for quantum-classical crossover in condensed matter problems''. npj Quantum Information 10, 45 (2024). doi: 10.1038/s41534-024-00839-4. https://doi.org/10.1038/s41534-024-00839-4 [12] Ali JavadiAbhari, Shruti Patil, Daniel Kudrow, Jeff Heckey, Alexey Lvov, Frederic T. Chong, and Margaret Martonosi. ``ScaffCC: A framework for compilation and analysis of quantum computing programs''. In Proceedings of the 11th ACM Conference on Computing Frontiers. CF '14. Association for Computing Machinery (2014). doi: 10.1145/2597917.2597939. https://doi.org/10.1145/2597917.2597939 [13] Qiskit contributors. ``Qiskit: An open-source framework for quantum computing''. url: doi.org/10.5281/zenodo.2573505. https://doi.org/10.5281/zenodo.2573505 [14] Seyon Sivarajah, Silas Dilkes, Alexander Cowtan, Will Simmons, Alec Edgington, and Ross Duncan. ``t|ket>: a retargetable compiler for NISQ devices''. Quantum Science and Technology 6, 014003 (2020). doi: 10.1088/2058-9565/ab8e92. https://doi.org/10.1088/2058-9565/ab8e92 [15] George Watkins, Hoang Minh Nguyen, Keelan Watkins, Steven Pearce, Hoi-Kwan Lau, and Alexandru Paler. ``A high performance compiler for very large scale surface code computations''. Quantum 8, 1354 (2024). doi: 10.22331/q-2024-05-22-1354. https://doi.org/10.22331/q-2024-05-22-1354 [16] Daniel Herr, Franco Nori, and Simon J. Devitt. ``Optimization of lattice surgery is np-hard''. Npj quantum information 3, 35 (2017). doi: 10.1038/s41534-017-0035-1. https://doi.org/10.1038/s41534-017-0035-1 [17] Abtin Molavi, Amanda Xu, Swamit Tannu, and Aws Albarghouthi. ``Compilation for surface code quantum computers'' (2023). doi: 10.48550/arXiv.2311.18042. arXiv:2311.18042. https://doi.org/10.48550/arXiv.2311.18042 arXiv:2311.18042 [18] Daniel Litinski. ``A game of surface codes: Large-scale quantum computing with lattice surgery''. Quantum 3, 128 (2019). doi: 10.22331/q-2019-03-05-128. https://doi.org/10.22331/q-2019-03-05-128 [19] Daniel Litinski. ``Magic state distillation: Not as costly as you think''. Quantum 3, 205 (2019). doi: 10.22331/q-2019-12-02-205. https://doi.org/10.22331/q-2019-12-02-205 [20] Lingling Lao, Bas van Wee, Imran Ashraf, J. van Someren, Nader Khammassi, Koen Bertels, and Carmen G. Almudever. ``Mapping of lattice surgery-based quantum circuits on surface code architectures''. Quantum Science and Technology 4, 015005 (2018). doi: 10.1088/2058-9565/aadd1a. https://doi.org/10.1088/2058-9565/aadd1a [21] Michael Beverland, Vadym Kliuchnikov, and Eddie Schoute. ``Surface code compilation via edge-disjoint paths''. PRX Quantum 3, 020342 (2022). doi: 10.1103/PRXQuantum.3.020342. https://doi.org/10.1103/PRXQuantum.3.020342 [22] Guang Hao Low and Isaac L. Chuang. ``Hamiltonian simulation by qubitization''. Quantum 3, 163 (2019). doi: 10.22331/q-2019-07-12-163. https://doi.org/10.22331/q-2019-07-12-163 [23] Austin G. Fowler, Adam C. Whiteside, and Lloyd C. L. Hollenberg. ``Towards practical classical processing for the surface code''. Physical review letters 108, 180501 (2012). doi: 10.1103/PhysRevLett.108.180501. https://doi.org/10.1103/PhysRevLett.108.180501 [24] Rajeev Acharya, Igor Aleiner, Richard Allen, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, et al. ``Suppressing quantum errors by scaling a surface code logical qubit''. Nature 614, 676–681 (2023). doi: 10.1038/s41586-022-05434-1. https://doi.org/10.1038/s41586-022-05434-1 [25] Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton. ``Poking holes and cutting corners to achieve clifford gates with the surface code''. Physical Review X 7, 021029 (2017). doi: 10.1103/PhysRevX.7.021029. https://doi.org/10.1103/PhysRevX.7.021029 [26] Michael L. Fredman and Robert Endre Tarjan. ``Fibonacci heaps and their uses in improved network optimization algorithms''. J. ACM 34, 596–615 (1987). doi: 10.1145/28869.28874. https://doi.org/10.1145/28869.28874 [27] Allyson Silva, Xiangyi Zhang, Zak Webb, Mia Kramer, Chan-Woo Yang, Xiao Liu, Jessica Lemieux, Ka-Wai Chen, Artur Scherer, and Pooya Ronagh. ``Multi-qubit lattice surgery scheduling''. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Pages 1–1. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2024). url: doi.org/10.4230/lipics.tqc.2024.1. https://doi.org/10.4230/lipics.tqc.2024.1 [28] Austin G. Fowler. ``Time-optimal quantum computation'' (2012). doi: 10.48550/arXiv.1210.4626. arXiv:1210.4626. https://doi.org/10.48550/arXiv.1210.4626 arXiv:1210.4626 [29] Mingzheng Zhu, Hao Fu, Jun Wu, Chi Zhang, Wei Xie, and Xiang-Yang Li. ``Ecmas: Efficient circuit mapping and scheduling for surface code''. In 2024 IEEE/ACM International Symposium on Code Generation and Optimization (CGO). Page 158–169. IEEE (2024). doi: 10.1109/cgo57630.2024.10444874. https://doi.org/10.1109/cgo57630.2024.10444874 [30] Mingzheng Zhu, Hao Fu, Haishan Song, Jun Wu, Chi Zhang, Wei Xie, and Xiangyang Li. ``Ecmas+: Efficient circuit mapping and scheduling for surface code encoded circuit on quantum cloud platform''. ACM Transactions on Architecture and Code Optimization 22, 1–25 (2025). doi: 10.1145/3760783. https://doi.org/10.1145/3760783 [31] Google Quantum AI. ``Suppressing quantum errors by scaling a surface code logical qubit''. Nature 614, 676–681 (2023). doi: 10.1038/s41586-022-05434-1. https://doi.org/10.1038/s41586-022-05434-1 [32] Google Quantum AI and Collaborators. ``Quantum error correction below the surface code threshold''. Nature 638, 920–926 (2024). doi: 10.1038/s41586-024-08449-y. https://doi.org/10.1038/s41586-024-08449-y [33] Hengyun Zhou, Chen Zhao, Madelyn Cain, Dolev Bluvstein, Nishad Maskara, Casey Duckering, Hong-Ye Hu, Sheng-Tao Wang, Aleksander Kubica, and Mikhail D. Lukin. ``Low-overhead transversal fault tolerance for universal quantum computation''. Nature 646, 303–308 (2025). doi: 10.1038/s41586-025-09543-5. https://doi.org/10.1038/s41586-025-09543-5 [34] Craig Gidney, Michael Newman, Peter Brooks, and Cody Jones. ``Yoked surface codes''. Nature Communications 16 (2025). doi: 10.1038/s41467-025-59714-1. https://doi.org/10.1038/s41467-025-59714-1 [35] Theodore J. Yoder, Eddie Schoute, Patrick Rall, Emily Pritchett, Jay M. Gambetta, Andrew W. Cross, Malcolm Carroll, and Michael E. Beverland. ``Tour de gross: A modular quantum computer based on bivariate bicycle codes'' (2025). doi: 10.48550/ARXIV.2506.03094. https://doi.org/10.48550/ARXIV.2506.03094 [36] Qian Xu, J.
Pablo Bonilla Ataides, Christopher A. Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasić, Mikhail D. Lukin, Liang Jiang, and Hengyun Zhou. ``Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays''. Nature Physics 20, 1084–1090 (2024). doi: 10.1038/s41567-024-02479-z. https://doi.org/10.1038/s41567-024-02479-z [37] Lawrence Z. Cohen, Isaac H. Kim, Stephen D. Bartlett, and Benjamin J. Brown. ``Low-overhead fault-tolerant quantum computing using long-range connectivity''. Science Advances 8 (2022). doi: 10.1126/sciadv.abn1717. https://doi.org/10.1126/sciadv.abn1717 [38] Naomi H Nickerson, Ying Li, and Simon C Benjamin. ``Topological quantum computing with a very noisy network and local error rates approaching one percent''. Nature communications 4, 1756 (2013). url: doi.org/10.1038/ncomms2773. https://doi.org/10.1038/ncomms2773 [39] Hector Bombin, Isaac H Kim, Daniel Litinski, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Sam Roberts, and Terry Rudolph. ``Interleaving: Modular architectures for fault-tolerant photonic quantum computing'' (2021). url: doi.org/10.48550/arXiv.2103.08612. https://doi.org/10.48550/arXiv.2103.08612 [40] J Eli Bourassa, Rafael N Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q Baragiola, Saikat Guha, Guillaume Dauphinais, et al. ``Blueprint for a scalable photonic fault-tolerant quantum computer''. Quantum 5, 392 (2021). url: doi.org/10.22331/q-2021-02-04-392. https://doi.org/10.22331/q-2021-02-04-392 [41] Christopher Pattison, Gefen Baranes, Juan Pablo Bonilla Ataides, Mikhail D Lukin, and Hengyun Zhou. ``Constant-rate entanglement distillation for fast quantum interconnects''. In Proceedings of the 52nd Annual International Symposium on Computer Architecture. Pages 257–270. (2025). url: doi.org/10.1145/3695053.373106. https://doi.org/10.1145/3695053.373106 [42] Yuya Maeda, Yasunari Suzuki, Toshiki Kobayashi, Takashi Yamamoto, Yuuki Tokunaga, and Keisuke Fujii. ``Logical entanglement distribution between distant 2d array qubits'' (2025). url: doi.org/10.48550/arXiv.2503.14894. https://doi.org/10.48550/arXiv.2503.14894 [43] Christopher Chamberland and Earl T Campbell. ``Universal quantum computing with twist-free and temporally encoded lattice surgery''. PRX Quantum 3, 010331 (2022). url: doi.org/10.1103/PRXQuantum.3.010331. https://doi.org/10.1103/PRXQuantum.3.010331 [44] Matt McEwen, Dave Bacon, and Craig Gidney. ``Relaxing hardware requirements for surface code circuits using time-dynamics''. Quantum 7, 1172 (2023). url: doi.org/10.22331/q-2023-11-07-1172. https://doi.org/10.22331/q-2023-11-07-1172 [45] Michael A Nielsen and Isaac L Chuang. ``Quantum computation and quantum information''. Cambridge university press. (2010). url: doi.org/10.1017/CBO9780511976667. https://doi.org/10.1017/CBO9780511976667 [46] Neil J Ross and Peter Selinger. ``Optimal ancilla-free clifford+ t approximation of z-rotations'' (2014). url: doi.org/10.48550/arXiv.1403.2975. https://doi.org/10.48550/arXiv.1403.2975 [47] David P DiVincenzo. ``Quantum gates and circuits''. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 261–276 (1998). url: doi.org/10.1098/rspa.1998.0159. https://doi.org/10.1098/rspa.1998.0159Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-13 17:48:12: Could not fetch cited-by data for 10.22331/q-2026-04-13-2061 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-13 17:48:13: 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.