The surface code with a twist

Theodore J. Yoder1 and Isaac H. Kim2

1Department of Physics, Massachusetts Institute of Technology
2IBM, Thomas J. Watson Research Center

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The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.

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[1] Sergey B Bravyi and A Yu Kitaev. Quantum codes on a lattice with boundary. quant-ph/​9811052, 1998.

[2] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43 (9): 4452–4505, 2002. 10.1063/​1.1499754.

[3] Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86 (3): 032324, 2012. 10.1103/​PhysRevA.86.032324.

[4] Dorit Aharonov and Lior Eldar. On the complexity of commuting local Hamiltonians, and tight conditions for topological order in such systems. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 334–343. IEEE, 2011. 10.1109/​FOCS.2011.58.

[5] Yu Tomita and Krysta M Svore. Low-distance surface codes under realistic quantum noise. Physical Review A, 90 (6): 062320, 2014. 10.1103/​PhysRevA.90.062320.

[6] David S Wang, Austin G Fowler, and Lloyd CL Hollenberg. Surface code quantum computing with error rates over 1%. Physical Review A, 83 (2): 020302, 2011. 10.1103/​PhysRevA.83.020302.

[7] Adrian Hutter, James R Wootton, and Daniel Loss. Efficient Markov chain Monte Carlo algorithm for the surface code. Physical Review A, 89 (2): 022326, 2014. 10.1103/​PhysRevA.89.022326.

[8] Sergey Bravyi, Martin Suchara, and Alexander Vargo. Efficient algorithms for maximum likelihood decoding in the surface code. Physical Review A, 90 (3): 032326, 2014. 10.1103/​PhysRevA.90.032326.

[9] James R Wootton and Daniel Loss. High threshold error correction for the surface code. Physical Review Letters, 109 (16): 160503, 2012. 10.1103/​PhysRevLett.109.160503.

[10] Austin G Fowler and Simon J Devitt. A bridge to lower overhead quantum computation. arXiv:1209.0510, 2012.

[11] Héctor Bombín. Topological subsystem codes. Physical Review A, 81 (3): 032301, 2010a. 10.1103/​PhysRevA.81.032301.

[12] Sergey Bravyi, Guillaume Duclos-Cianci, David Poulin, and Martin Suchara. Subsystem surface codes with three-qubit check operators. Quantum Information & Computation, 13 (11-12): 963–985, 2013.

[13] Héctor Bombín and Miguel Angel Martin-Delgado. Topological quantum distillation. Physical Review Letters, 97 (18): 180501, 2006. 10.1103/​PhysRevLett.97.180501.

[14] Héctor Bombín. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New Journal of Physics, 17 (8): 083002, 2015. 10.1088/​1367-2630/​17/​8/​083002.

[15] Aleksander Kubica and Michael E Beverland. Universal transversal gates with color codes: A simplified approach. Physical Review A, 91 (3): 032330, 2015. 10.1103/​PhysRevA.91.032330.

[16] Andrew J Landahl and Ciaran Ryan-Anderson. Quantum computing by color-code lattice surgery. arXiv:1407.5103, 2014.

[17] Robert Raussendorf and Jim Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Physical Review Letters, 98 (19): 190504, 2007. 10.1103/​PhysRevLett.98.190504.

[18] H Bombin and MA Martin-Delgado. Quantum measurements and gates by code deformation. Journal of Physics A: Mathematical and Theoretical, 42 (9): 095302, 2009. 10.1088/​1751-8113/​42/​9/​095302.

[19] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Physical Review A, 71 (2): 022316, 2005. 10.1103/​PhysRevA.71.022316.

[20] Austin G Fowler, Simon J Devitt, and Cody Jones. Surface code implementation of block code state distillation. Scientific reports, 3, 2013. 10.1038/​srep01939.

[21] Austin G Fowler, Ashley M Stephens, and Peter Groszkowski. High-threshold universal quantum computation on the surface code. Physical Review A, 80 (5): 052312, 2009. 10.1103/​PhysRevA.80.052312.

[22] Austin G Fowler. Time-optimal quantum computation. arXiv:1210.4626, 2012.

[23] Clare Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. Surface code quantum computing by lattice surgery. New Journal of Physics, 14 (12): 123011, 2012. 10.1088/​1367-2630/​14/​12/​123011.

[24] Héctor Bombín. Topological order with a twist: Ising anyons from an Abelian model. Physical Review Letters, 105 (3): 030403, 2010b. 10.1103/​PhysRevLett.105.030403.

[25] Matthew B Hastings and A Geller. Reduced space-time and time costs using dislocation codes and arbitrary ancillas. Quantum Information & Computation, 15 (11-12): 962–986, 2015.

[26] A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54: 1098–1105, 1996. 10.1103/​PhysRevA.54.1098.

[27] Andrew M Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77 (5): 793, 1996. 10.1103/​PhysRevLett.77.793.

[28] 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. arXiv:1609.04673, 2016.

[29] H Bombin and MA Martin-Delgado. Optimal resources for topological two-dimensional stabilizer codes: Comparative study. Physical Review A, 76 (1): 012305, 2007. 10.1103/​PhysRevA.76.012305.

[30] Daniel Gottesman and Isaac L Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402 (6760): 390–393, 1999. 10.1038/​46503.

[31] Peter W Shor. Fault-tolerant quantum computation. In Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on, pages 56–65. IEEE, 1996. 10.1109/​SFCS.1996.548464.

[32] Peter Brooks and John Preskill. Fault-tolerant quantum computation with asymmetric Bacon-Shor codes. Physical Review A, 87 (3): 032310, 2013. 10.1103/​PhysRevA.87.032310.

[33] Jonathan E Moussa. Transversal clifford gates on folded surface codes. Physical Review A, 94 (4): 042316, 2016. 10.1103/​PhysRevA.94.042316.

[34] Aleksander Kubica, Beni Yoshida, and Fernando Pastawski. Unfolding the color code. New Journal of Physics, 17 (8): 083026, 2015. 10.1088/​1367-2630/​17/​8/​083026.

[35] Andrew W Cross. personal communication.

[36] Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Information & Computation, 6 (2): 97–165, 2006.

[37] Krysta M Svore, Andrew W Cross, Isaac L Chuang, and Alfred V Aho. A flow-map model for analyzing pseudothresholds in fault-tolerant quantum computing. Quantum Information & Computation, 6 (3): 193–212, 2006.

[38] Daniel Gottesman. Quantum fault tolerance in small experiments. arXiv preprint arXiv:1610.03507, 2016.

[39] Andrew W Cross, David P Divincenzo, and Barbara M Terhal. A comparative code study for quantum fault tolerance. Quantum Information & Computation, 9 (7): 541–572, 2009.

[40] Emanuel Knill. Quantum computing with realistically noisy devices. Nature, 434 (7029): 39–44, 2005. 10.1038/​nature03350.

[41] Charles H Bennett, David P DiVincenzo, John A Smolin, and William K Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54 (5): 3824, 1996. 10.1103/​PhysRevA.54.3824.

[42] Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. Perfect quantum error correcting code. Physical Review Letters, 77 (1): 198, 1996. 10.1103/​PhysRevLett.77.198.

[43] David P DiVincenzo and Panos Aliferis. Effective fault-tolerant quantum computation with slow measurements. Physical Review Letters, 98 (2): 020501, 2007. 10.1103/​PhysRevLett.98.020501.

[44] Ashley M Stephens. Efficient fault-tolerant decoding of topological color codes. arXiv preprint arXiv:1402.3037, 2014.

[45] Panos Aliferis and Andrew W Cross. Subsystem fault tolerance with the Bacon-Shor code. Physical Review Letters, 98 (22): 220502, 2007. 10.1103/​PhysRevLett.98.220502.

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[2] Héctor Bombín, Mihir Pant, Sam Roberts, and Karthik I. Seetharam, "Fault-Tolerant Postselection for Low-Overhead Magic State Preparation", PRX Quantum 5 1, 010302 (2024).

[3] Daniel Herr, Alexandru Paler, Simon J Devitt, and Franco Nori, "Lattice surgery on the Raussendorf lattice", Quantum Science and Technology 3 3, 035011 (2018).

[4] Kao-Yueh Kuo and Ching-Yi Lai, 2022 IEEE International Symposium on Information Theory (ISIT) 186 (2022) ISBN:978-1-6654-2159-1.

[5] Anirudh Krishna and David Poulin, "Fault-Tolerant Gates on Hypergraph Product Codes", Physical Review X 11 1, 011023 (2021).

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[7] J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown, "The XZZX surface code", Nature Communications 12 1, 2172 (2021).

[8] Christopher Chamberland and Earl T. Campbell, "Circuit-level protocol and analysis for twist-based lattice surgery", Physical Review Research 4 2, 023090 (2022).

[9] Sam Roberts and Dominic J. Williamson, "3-Fermion Topological Quantum Computation", PRX Quantum 5 1, 010315 (2024).

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

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

[12] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, "Quantum Chemistry in the Age of Quantum Computing", Chemical Reviews 119 19, 10856 (2019).

[13] Dhruv Bhatnagar, Matthew Steinberg, David Elkouss, Carmen G. Almudever, and Sebastian Feld, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 63 (2023) ISBN:979-8-3503-4323-6.

[14] Laxmidhar Biswal, Anirban Bhattacharjee, Rakesh Das, Gopinath Thirunavukarasu, and Hafizur Rahaman, Communications in Computer and Information Science 892, 606 (2019) ISBN:978-981-13-5949-1.

[15] Markus S. Kesselring, Julio C. Magdalena de la Fuente, Felix Thomsen, Jens Eisert, Stephen D. Bartlett, and Benjamin J. Brown, "Anyon Condensation and the Color Code", PRX Quantum 5 1, 010342 (2024).

[16] Michael E Beverland, Benjamin J Brown, Michael J Kastoryano, and Quentin Marolleau, "The role of entropy in topological quantum error correction", Journal of Statistical Mechanics: Theory and Experiment 2019 7, 073404 (2019).

[17] Theerapat Tansuwannont and Debbie Leung, "Fault-tolerant quantum error correction using error weight parities", Physical Review A 104 4, 042410 (2021).

[18] Eduardo Brandani da Silva and Waldir S Soares, "Construction of color codes from polygons", Journal of Physics Communications 2 9, 095011 (2018).

[19] Corey Ostrove, Brian La Cour, Andrew Lanham, and Granville Ott, "Improving performance of an analog electronic device using quantum error correction", Journal of Physics Communications 3 8, 085017 (2019).

[20] Nishad Maskara, Aleksander Kubica, and Tomas Jochym-O'Connor, "Advantages of versatile neural-network decoding for topological codes", Physical Review A 99 5, 052351 (2019).

[21] Kianna Wan, Soonwon Choi, Isaac H. Kim, Noah Shutty, and Patrick Hayden, "Fault-Tolerant Qubit from a Constant Number of Components", PRX Quantum 2 4, 040345 (2021).

[22] Jhih-Yuan Kao and Hsi-Sheng Goan, "Existence of Pauli-like stabilizers for every quantum error-correcting code", Physical Review A 108 3, 032414 (2023).

[23] Iris Cong, Harry Levine, Alexander Keesling, Dolev Bluvstein, Sheng-Tao Wang, and Mikhail D. Lukin, "Hardware-Efficient, Fault-Tolerant Quantum Computation with Rydberg Atoms", Physical Review X 12 2, 021049 (2022).

[24] Alexander Zlokapa, Benjamin Villalonga, Sergio Boixo, and Daniel A. Lidar, "Boundaries of quantum supremacy via random circuit sampling", npj Quantum Information 9 1, 36 (2023).

[25] Lingling Lao and Carmen G. Almudever, "Fault-tolerant quantum error correction on near-term quantum processors using flag and bridge qubits", Physical Review A 101 3, 032333 (2020).

[26] 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", Physical Review X 10 1, 011022 (2020).

[27] Christopher Chamberland and Earl T. Campbell, "Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery", PRX Quantum 3 1, 010331 (2022).

[28] Yujin Kang, Jonghyun Lee, Jinyoung Ha, and Jun Heo, "Fault-tolerant quantum computation using low-cost joint measurements", Quantum Information Processing 23 5, 190 (2024).

[29] Ki-Sung Jin and Gyu-Il Cha, "Multilayered logical qubits and synthesized quantum bits", Quantum Science and Technology 8 3, 035008 (2023).

[30] Daniel Litinski and Felix von Oppen, "Braiding by Majorana tracking and long-range CNOT gates with color codes", Physical Review B 96 20, 205413 (2017).

[31] Noah Shutty and Christopher Chamberland, "Decoding Merged Color-Surface Codes and Finding Fault-Tolerant Clifford Circuits Using Solvers for Satisfiability Modulo Theories", Physical Review Applied 18 1, 014072 (2022).

[32] Markus S. Kesselring, Fernando Pastawski, Jens Eisert, and Benjamin J. Brown, "The boundaries and twist defects of the color code and their applications to topological quantum computation", Quantum 2, 101 (2018).

[33] Laxmidhar Biswal, Rakesh Das, Chandan Bandyopadhyay, Anupam Chattopadhyay, and Hafizur Rahaman, "A template-based technique for efficient Clifford+T-based quantum circuit implementation", Microelectronics Journal 81, 58 (2018).

[34] Jing Hao Chai and Hui Khoon Ng, "On the Fault‐Tolerance Threshold for Surface Codes with General Noise", Advanced Quantum Technologies 5 10, 2200008 (2022).

[35] Ryuji Takagi, Theodore J. Yoder, and Isaac L. Chuang, "Error rates and resource overheads of encoded three-qubit gates", Physical Review A 96 4, 042302 (2017).

[36] Christopher Chamberland and Kyungjoo Noh, "Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits", npj Quantum Information 6 1, 91 (2020).

[37] Rui Chao and Ben W. Reichardt, "Quantum Error Correction with Only Two Extra Qubits", Physical Review Letters 121 5, 050502 (2018).

[38] Dripto M Debroy, Muyuan Li, Shilin Huang, and Kenneth R Brown, "Logical performance of 9 qubit compass codes in ion traps with crosstalk errors", Quantum Science and Technology 5 3, 034002 (2020).

[39] Colin J Trout, Muyuan Li, Mauricio Gutiérrez, Yukai Wu, Sheng-Tao Wang, Luming Duan, and Kenneth R Brown, "Simulating the performance of a distance-3 surface code in a linear ion trap", New Journal of Physics 20 4, 043038 (2018).

[40] Manoj G. Gowda and Pradeep Kiran Sarvepalli, "Quantum computation with charge-and-color-permuting twists in qudit color codes", Physical Review A 105 2, 022621 (2022).

[41] Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, Fernando Pastawski, and Sam Roberts, "Logical Blocks for Fault-Tolerant Topological Quantum Computation", PRX Quantum 4 2, 020303 (2023).

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[43] Daniel C. Murphy and Kenneth R. Brown, "Controlling error orientation to improve quantum algorithm success rates", Physical Review A 99 3, 032318 (2019).

[44] Dripto M. Debroy and Kenneth R. Brown, "Extended flag gadgets for low-overhead circuit verification", Physical Review A 102 5, 052409 (2020).

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[46] 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 2, 021029 (2017).

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[49] Christopher Chamberland and Michael E. Beverland, "Flag fault-tolerant error correction with arbitrary distance codes", Quantum 2, 53 (2018).

[50] Ben W Reichardt, "Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits", Quantum Science and Technology 6 1, 015007 (2021).

[51] E. J. Crosson and D. A. Lidar, "Prospects for quantum enhancement with diabatic quantum annealing", Nature Reviews Physics 3 7, 466 (2021).

[52] Pei-Hao Liou and Ching-Yi Lai, "Parallel syndrome extraction with shared flag qubits for Calderbank-Shor-Steane codes of distance three", Physical Review A 107 2, 022614 (2023).

[53] Muyuan Li, Daniel Miller, and Kenneth R. Brown, "Direct measurement of Bacon-Shor code stabilizers", Physical Review A 98 5, 050301 (2018).

[54] Prithviraj Prabhu and Ben W. Reichardt, "Fault-tolerant syndrome extraction and cat state preparation with fewer qubits", Quantum 7, 1154 (2023).

[55] X. Fu, L. Lao, K. Bertels, and C.G. Almudever, "A control microarchitecture for fault-tolerant quantum computing", Microprocessors and Microsystems 70, 21 (2019).

[56] Christopher Chamberland and Pooya Ronagh, "Deep neural decoders for near term fault-tolerant experiments", Quantum Science and Technology 3 4, 044002 (2018).

[57] Alexis Schotte, Guanyu Zhu, Lander Burgelman, and Frank Verstraete, "Quantum Error Correction Thresholds for the Universal Fibonacci Turaev-Viro Code", Physical Review X 12 2, 021012 (2022).

[58] Anirudh Krishna and David Poulin, "Topological wormholes: Nonlocal defects on the toric code", Physical Review Research 2 2, 023116 (2020).

[59] Antonio deMarti iOlius, Josu Etxezarreta Martinez, Patricio Fuentes, Pedro M. Crespo, and Javier Garcia-Frias, "Performance of surface codes in realistic quantum hardware", Physical Review A 106 6, 062428 (2022).

[60] Shilin Huang, Tomas Jochym-O’Connor, and Theodore J. Yoder, "Homomorphic Logical Measurements", PRX Quantum 4 3, 030301 (2023).

[61] Paul Webster and Stephen D. Bartlett, "Fault-tolerant quantum gates with defects in topological stabilizer codes", Physical Review A 102 2, 022403 (2020).

[62] Daniel Litinski and Felix von Oppen, "Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes", Quantum 2, 62 (2018).

[63] Daniel Litinski and Felix von Oppen, "Quantum computing with Majorana fermion codes", Physical Review B 97 20, 205404 (2018).

[64] Muyuan Li, Daniel Miller, Michael Newman, Yukai Wu, and Kenneth R. Brown, "2D Compass Codes", Physical Review X 9 2, 021041 (2019).

[65] Tomas Jochym-O'Connor and Theodore J. Yoder, "Four-dimensional toric code with non-Clifford transversal gates", Physical Review Research 3 1, 013118 (2021).

[66] Ali Lavasani and Maissam Barkeshli, "Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes", Physical Review A 98 5, 052319 (2018).

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

[68] 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", Physical Review Letters 119 18, 180501 (2017).

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[70] Muyuan Li, Mauricio Gutiérrez, Stanley E. David, Alonzo Hernandez, and Kenneth R. Brown, "Fault tolerance with bare ancillary qubits for a [[7,1,3]] code", Physical Review A 96 3, 032341 (2017).

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[72] QiFei Wei, DongXiao Quan, Jing Liu, and ChangXing Pei, 2019 20th International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT) 173 (2019) ISBN:978-1-7281-2616-6.

[73] Christopher Chamberland and Andrew W. Cross, "Fault-tolerant magic state preparation with flag qubits", Quantum 3, 143 (2019).

[74] Natalie C Brown, Michael Newman, and Kenneth R Brown, "Handling leakage with subsystem codes", New Journal of Physics 21 7, 073055 (2019).

[75] Benjamin J. Brown and Sam Roberts, "Universal fault-tolerant measurement-based quantum computation", Physical Review Research 2 3, 033305 (2020).

[76] 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", arXiv:2310.03011, (2023).

[77] T. E. O'Brien, B. Tarasinski, and L. DiCarlo, "Density-matrix simulation of small surface codes under current and projected experimental noise", npj Quantum Information 3, 39 (2017).

[78] ChunJun Cao and Brad Lackey, "Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks", PRX Quantum 3 2, 020332 (2022).

[79] Remmy Zen, Jan Olle, Luis Colmenarez, Matteo Puviani, Markus Müller, and Florian Marquardt, "Quantum Circuit Discovery for Fault-Tolerant Logical State Preparation with Reinforcement Learning", arXiv:2402.17761, (2024).

[80] Andrew J. Landahl and Benjamin C. A. Morrison, "Logical fermions for fault-tolerant quantum simulation", arXiv:2110.10280, (2021).

[81] Ben W. Reichardt, "Fault-tolerant quantum error correction for Steane's seven-qubit color code with few or no extra qubits", arXiv:1804.06995, (2018).

[82] Nishad Maskara, Aleksander Kubica, and Tomas Jochym-O'Connor, "Advantages of versatile neural-network decoding for topological codes", arXiv:1802.08680, (2018).

[83] Samuel J. Elman, Jason Gavriel, and Ryan L. Mann, "Optimal Scheduling of Graph States via Path Decompositions", arXiv:2403.04126, (2024).

[84] Dhruv Bhatnagar, Matthew Steinberg, David Elkouss, Carmen G. Almudever, and Sebastian Feld, "Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes", arXiv:2305.00784, (2023).

[85] Andrew J. Landahl, "The surface code on the rhombic dodecahedron", arXiv:2010.06628, (2020).

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