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Data-driven decoding of quantum error correcting codes using graph neural networks (2307.01241v2)

Published 3 Jul 2023 in quant-ph

Abstract: To leverage the full potential of quantum error-correcting stabilizer codes it is crucial to have an efficient and accurate decoder. Accurate, maximum likelihood, decoders are computationally very expensive whereas decoders based on more efficient algorithms give sub-optimal performance. In addition, the accuracy will depend on the quality of models and estimates of error rates for idling qubits, gates, measurements, and resets, and will typically assume symmetric error channels. In this work, instead, we explore a model-free, data-driven, approach to decoding, using a graph neural network (GNN). The decoding problem is formulated as a graph classification task in which a set of stabilizer measurements is mapped to an annotated detector graph for which the neural network predicts the most likely logical error class. We show that the GNN-based decoder can outperform a matching decoder for circuit level noise on the surface code given only simulated experimental data, even if the matching decoder is given full information of the underlying error model. Although training is computationally demanding, inference is fast and scales approximately linearly with the space-time volume of the code. We also find that we can use large, but more limited, datasets of real experimental data [Google Quantum AI, Nature {\bf 614}, 676 (2023)] for the repetition code, giving decoding accuracies that are on par with minimum weight perfect matching. The results show that a purely data-driven approach to decoding may be a viable future option for practical quantum error correction, which is competitive in terms of speed, accuracy, and versatility.

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References (112)
  1. Peter W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Physical Review A 52, R2493–R2496 (1995).
  2. A. M. Steane, “Error Correcting Codes in Quantum Theory,” Physical Review Letters 77, 793–797 (1996).
  3. Daniel Gottesman, “Stabilizer Codes and Quantum Error Correction,”  (1997), arXiv:quant-ph/9705052 .
  4. Barbara M. Terhal, “Quantum error correction for quantum memories,” Reviews of Modern Physics 87, 307–346 (2015).
  5. Steven M. Girvin, “Introduction to quantum error correction and fault tolerance,”  (2021), arXiv:2111.08894 .
  6. 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).
  7. Kristan Temme, Sergey Bravyi,  and Jay M Gambetta, “Error mitigation for short-depth quantum circuits,” Physical review letters 119, 180509 (2017).
  8. Ying Li and Simon C Benjamin, “Efficient variational quantum simulator incorporating active error minimization,” Physical Review X 7, 021050 (2017).
  9. S. B. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary,”  (1998), arXiv:quant-ph/9811052 .
  10. Eric Dennis, Alexei Kitaev, Andrew Landahl,  and John Preskill, “Topological quantum memory,” Journal of Mathematical Physics 43, 4452–4505 (2002a).
  11. A.Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2–30 (2003).
  12. Robert Raussendorf and Jim Harrington, “Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions,” Physical Review Letters 98, 190504 (2007).
  13. 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).
  14. J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland,  and John M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature 519, 66–69 (2015).
  15. 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, 180501 (2017).
  16. James R. Wootton and Daniel Loss, “Repetition code of 15 qubits,” Phys. Rev. A 97, 052313 (2018).
  17. James R Wootton, “Benchmarking near-term devices with quantum error correction,” Quantum Science and Technology 5, 044004 (2020).
  18. Christian Kraglund Andersen, Ants Remm, Stefania Lazar, Sebastian Krinner, Nathan Lacroix, Graham J. Norris, Mihai Gabureac, Christopher Eichler,  and Andreas Wallraff, “Repeated quantum error detection in a surface code,” Nature Physics 16, 875–880 (2020).
  19. K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor,” Science 374, 1237 (2021).
  20. Laird Egan, Dripto M. Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael Newman, Muyuan Li, Kenneth R. Brown, Marko Cetina,  and Christopher Monroe, “Fault-tolerant control of an error-corrected qubit,” Nature 598, 281–286 (2021).
  21. Zijun Chen et al., “Exponential suppression of bit or phase errors with cyclic error correction,” Nature 595, 383–387 (2021).
  22. Alexander Erhard, Hendrik Poulsen Nautrup, Michael Meth, Lukas Postler, Roman Stricker, Martin Stadler, Vlad Negnevitsky, Martin Ringbauer, Philipp Schindler, Hans J. Briegel, Rainer Blatt, Nicolai Friis,  and Thomas Monz, “Entangling logical qubits with lattice surgery,” Nature 589, 220–224 (2021).
  23. 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).
  24. J. F. Marques, B. M. Varbanov, M. S. Moreira, H. Ali, N. Muthusubramanian, C. Zachariadis, F. Battistel, M. Beekman, N. Haider, W. Vlothuizen, A. Bruno, B. M. Terhal,  and L. DiCarlo, “Logical-qubit operations in an error-detecting surface code,” Nature Physics 18, 80–86 (2021).
  25. Lukas Postler, Sascha Heussen, Ivan Pogorelov, Manuel Rispler, Thomas Feldker, Michael Meth, Christian D. Marciniak, Roman Stricker, Martin Ringbauer, Rainer Blatt, Philipp Schindler, Markus Müller,  and Thomas Monz, “Demonstration of fault-tolerant universal quantum gate operations,” Nature 605, 675–680 (2022).
  26. Sebastian Krinner, Nathan Lacroix, Ants Remm, Agustin Di Paolo, Elie Genois, Catherine Leroux, Christoph Hellings, Stefania Lazar, Christian Kraglund Andersen, et al., “Realizing repeated quantum error correction in a distance-three surface code,” Nature 605, 669–674 (2022).
  27. 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).
  28. Google Quantum AI, “Suppressing quantum errors by scaling a surface code logical qubit,” Nature 614, 676–681 (2023).
  29. S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn, J. G. Bohnet, N. C. Brown, N. Q. Burdick, W. C. Burton, S. L. Campbell, J. P. Campora III au2, C. Carron, J. Chambers, J. W. Chan, Y. H. Chen, A. Chernoguzov, E. Chertkov, J. Colina, J. P. Curtis, R. Daniel, M. DeCross, D. Deen, C. Delaney, J. M. Dreiling, C. T. Ertsgaard, J. Esposito, B. Estey, M. Fabrikant, C. Figgatt, C. Foltz, M. Foss-Feig, D. Francois, J. P. Gaebler, T. M. Gatterman, C. N. Gilbreth, J. Giles, E. Glynn, A. Hall, A. M. Hankin, A. Hansen, D. Hayes, B. Higashi, I. M. Hoffman, B. Horning, J. J. Hout, R. Jacobs, J. Johansen, L. Jones, J. Karcz, T. Klein, P. Lauria, P. Lee, D. Liefer, C. Lytle, S. T. Lu, D. Lucchetti, A. Malm, M. Matheny, B. Mathewson, K. Mayer, D. B. Miller, M. Mills, B. Neyenhuis, L. Nugent, S. Olson, J. Parks, G. N. Price, Z. Price, M. Pugh, A. Ransford, A. P. Reed, C. Roman, M. Rowe, C. Ryan-Anderson, S. Sanders, J. Sedlacek, P. Shevchuk, P. Siegfried, T. Skripka, B. Spaun, R. T. Sprenkle, R. P. Stutz, M. Swallows, R. I. Tobey, A. Tran, T. Tran, E. Vogt, C. Volin, J. Walker, A. M. Zolot,  and J. M. Pino, “A race track trapped-ion quantum processor,”  (2023), arXiv:2305.03828 [quant-ph] .
  30. Neereja Sundaresan, Theodore J Yoder, Youngseok Kim, Muyuan Li, Edward H Chen, Grace Harper, Ted Thorbeck, Andrew W Cross, Antonio D Córcoles,  and Maika Takita, “Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders,” Nature Communications 14, 2852 (2023).
  31. Craig Gidney, “Stim: a fast stabilizer circuit simulator,” Quantum 5, 497 (2021).
  32. Oscar Higgott, “Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching,” arXiv preprint arXiv:2105.13082  (2021).
  33. Emanuel Knill, “Quantum computing with realistically noisy devices,” Nature 434, 39–44 (2005).
  34. Niel de Beaudrap and Dominic Horsman, “The ZX calculus is a language for surface code lattice surgery,” Quantum 4, 218 (2020).
  35. James R. Wootton and Daniel Loss, “High Threshold Error Correction for the Surface Code,” Physical Review Letters 109, 160503 (2012).
  36. Adrian Hutter, James R. Wootton,  and Daniel Loss, “Efficient Markov chain Monte Carlo algorithm for the surface code,” Physical Review A 89, 022326 (2014).
  37. Sergey Bravyi, Martin Suchara,  and Alexander Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code,” Physical Review A 90, 032326 (2014).
  38. Karl Hammar, Alexei Orekhov, Patrik Wallin Hybelius, Anna Katariina Wisakanto, Basudha Srivastava, Anton Frisk Kockum,  and Mats Granath, “Error-rate-agnostic decoding of topological stabilizer codes,” Phys. Rev. A 105, 042616 (2022).
  39. Leonid P Pryadko, “On maximum-likelihood decoding with circuit-level errors,” Quantum 4, 304 (2020).
  40. Christopher T. Chubb, “General tensor network decoding of 2d pauli codes,”  (2021).
  41. Jack Edmonds, “Paths, trees, and flowers,” Canadian Journal of Mathematics 17, 449 (1965).
  42. Eric Dennis, Alexei Kitaev, Andrew Landahl,  and John Preskill, “Topological quantum memory,” Journal of Mathematical Physics 43, 4452 (2002b).
  43. D. S. Wang, A. G. Fowler, A. M. Stephens,  and L. C. L. Hollenberg, “Threshold Error Rates for the Toric and Planar Codes,” Quantum Information & Computation 10, 456 (2010).
  44. David S. Wang, Austin G. Fowler,  and Lloyd C. L. Hollenberg, “Surface code quantum computing with error rates over 1%,” Physical Review A 83, 020302 (2011).
  45. Austin G Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time,” Quantum Information and Computation 15, 145 (2015).
  46. Benjamin J. Brown, “Conservation laws and quantum error correction: towards a generalised matching decoder,”  (2022).
  47. Nicolas Delfosse, “Decoding color codes by projection onto surface codes,” Physical Review A 89 (2014), 10.1103/physreva.89.012317.
  48. Ashley M. Stephens, “Efficient fault-tolerant decoding of topological color codes,”  (2014).
  49. Nicolas Delfosse and Naomi H. Nickerson, “Almost-linear time decoding algorithm for topological codes,” Quantum 5, 595 (2021).
  50. David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia,  and Benjamin J. Brown, “Fault-tolerant thresholds for the surface code in excess of 5% under biased noise,” Physical Review Letters 124 (2020), 10.1103/physrevlett.124.130501.
  51. Kaavya Sahay and Benjamin J. Brown, ‘‘Decoder for the triangular color code by matching on a möbius strip,” PRX Quantum 3 (2022), 10.1103/prxquantum.3.010310.
  52. Lucas Berent, Lukas Burgholzer, Peter-Jan H. S. Derks, Jens Eisert,  and Robert Wille, “Decoding quantum color codes with maxsat,”  (2023).
  53. Asmae Benhemou, Kaavya Sahay, Lingling Lao,  and Benjamin J. Brown, “Minimising surface-code failures using a color-code decoder,”  (2023).
  54. Nicolas Delfosse and Jean-Pierre Tillich, “A decoding algorithm for CSS codes using the X/Z correlations,” in 2014 IEEE International Symposium on Information Theory (IEEE, 2014) pp. 1071–1075.
  55. Ben Criger and Imran Ashraf, “Multi-path summation for decoding 2D topological codes,” Quantum 2, 102 (2018).
  56. Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia,  and Earl T. Campbell, “Fragile boundaries of tailored surface codes and improved decoding of circuit-level noise,”  (2022).
  57. Laura Caune, Joan Camps, Brendan Reid,  and Earl Campbell, “Belief propagation as a partial decoder,”  (2023).
  58. Michael Herold, Earl T Campbell, Jens Eisert,  and Michael J Kastoryano, “Cellular-automaton decoders for topological quantum memories,” npj Quantum Information 1, 15010 (2015).
  59. Aleksander Kubica and John Preskill, “Cellular-Automaton Decoders with Provable Thresholds for Topological Codes,” Physical Review Letters 123, 020501 (2019).
  60. Jonathan F. San Miguel, Dominic J. Williamson,  and Benjamin J. Brown, “A cellular automaton decoder for a noise-bias tailored color code,” Quantum 7, 940 (2023).
  61. Guillaume Duclos-Cianci and David Poulin, “Fast Decoders for Topological Quantum Codes,” Physical Review Letters 104, 050504 (2010).
  62. Shilin Huang, Michael Newman,  and Kenneth R. Brown, “Fault-tolerant weighted union-find decoding on the toric code,” Physical Review A 102, 012419 (2020).
  63. Giacomo Torlai and Roger G. Melko, “Neural Decoder for Topological Codes,” Physical Review Letters 119, 030501 (2017).
  64. Stefan Krastanov and Liang Jiang, “Deep neural network probabilistic decoder for stabilizer codes,” Scientific Reports 7, 11003 (2017).
  65. Savvas Varsamopoulos, Ben Criger,  and Koen Bertels, “Decoding small surface codes with feedforward neural networks,” Quantum Science and Technology 3, 015004 (2017).
  66. Paul Baireuther, Thomas E O’Brien, Brian Tarasinski,  and Carlo WJ Beenakker, “Machine-learning-assisted correction of correlated qubit errors in a topological code,” Quantum 2, 48 (2018).
  67. Nikolas P Breuckmann and Xiaotong Ni, “Scalable Neural Network Decoders for Higher Dimensional Quantum Codes,” Quantum 2, 68 (2018).
  68. P Baireuther, M D Caio, B Criger, C W J Beenakker,  and T E O’Brien, “Neural network decoder for topological color codes with circuit level noise,” New Journal of Physics 21, 013003 (2019).
  69. Christopher Chamberland and Pooya Ronagh, “Deep neural decoders for near term fault-tolerant experiments,” Quantum Science and Technology 3, 044002 (2018).
  70. Hendrik Poulsen Nautrup, Nicolas Delfosse, Vedran Dunjko, Hans J Briegel,  and Nicolai Friis, “Optimizing Quantum Error Correction Codes with Reinforcement Learning,” Quantum 3, 215 (2019).
  71. Nishad Maskara, Aleksander Kubica,  and Tomas Jochym-O’Connor, “Advantages of versatile neural-network decoding for topological codes,” Physical Review A 99, 052351 (2019).
  72. Xiaotong Ni, “Neural Network Decoders for Large-Distance 2D Toric Codes,” Quantum 4, 310 (2020).
  73. Philip Andreasson, Joel Johansson, Simon Liljestrand,  and Mats Granath, “Quantum error correction for the toric code using deep reinforcement learning,” Quantum 3, 183 (2019).
  74. David Fitzek, Mattias Eliasson, Anton Frisk Kockum,  and Mats Granath, “Deep Q-learning decoder for depolarizing noise on the toric code,” Physical Review Research 2, 023230 (2020).
  75. Debasmita Bhoumik, Pinaki Sen, Ritajit Majumdar, Susmita Sur-Kolay, Latesh Kumar K J,  and Sundaraja Sitharama Iyengar, “Efficient decoding of surface code syndromes for error correction in quantum computing,”  (2021), arXiv:2110.10896 [quant-ph] .
  76. Hugo Théveniaut and Evert van Nieuwenburg, ‘‘A NEAT Quantum Error Decoder,” SciPost Physics 11, 5 (2021).
  77. Kai Meinerz, Chae-Yeun Park,  and Simon Trebst, “Scalable neural decoder for topological surface codes,” Physical Review Letters 128 (2022), 10.1103/physrevlett.128.080505.
  78. Christopher Chamberland, Luis Goncalves, Prasahnt Sivarajah, Eric Peterson,  and Sebastian Grimberg, “Techniques for combining fast local decoders with global decoders under circuit-level noise,”  (2022), arXiv:2208.01178 [quant-ph] .
  79. Mengyu Zhang, Xiangyu Ren, Guanglei Xi, Zhenxing Zhang, Qiaonian Yu, Fuming Liu, Hualiang Zhang, Shengyu Zhang,  and Yi-Cong Zheng, “A scalable, fast and programmable neural decoder for fault-tolerant quantum computation using surface codes,”  (2023), arXiv:2305.15767 [quant-ph] .
  80. Thomas Wagner, Hermann Kampermann, Dagmar Bruß,  and Martin Kliesch, “Pauli channels can be estimated from syndrome measurements in quantum error correction,” Quantum 6, 809 (2022).
  81. Edward H. Chen, Theodore J. Yoder, Youngseok Kim, Neereja Sundaresan, Srikanth Srinivasan, Muyuan Li, Antonio D. Córcoles, Andrew W. Cross,  and Maika Takita, “Calibrated decoders for experimental quantum error correction,” Phys. Rev. Lett. 128, 110504 (2022).
  82. Thomas Wagner, Hermann Kampermann, Dagmar Bruß,  and Martin Kliesch, “Learning logical pauli noise in quantum error correction,” Phys. Rev. Lett. 130, 200601 (2023).
  83. H. Bombin and M. A. Martin-Delgado, “Optimal resources for topological two-dimensional stabilizer codes: Comparative study,” Physical Review A 76, 012305 (2007).
  84. Yu Tomita and Krysta M. Svore, ‘‘Low-distance surface codes under realistic quantum noise,” Physical Review A 90, 062320 (2014).
  85. David K. Tuckett, Andrew S. Darmawan, Christopher T. Chubb, Sergey Bravyi, Stephen D. Bartlett,  and Steven T. Flammia, “Tailoring Surface Codes for Highly Biased Noise,” Physical Review X 9, 041031 (2019).
  86. Thomas N Kipf and Max Welling, “Semi-supervised classification with graph convolutional networks,” arXiv preprint arXiv:1609.02907  (2016).
  87. Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang,  and S Yu Philip, “A comprehensive survey on graph neural networks,” IEEE transactions on neural networks and learning systems  (2020).
  88. Vijay Prakash Dwivedi, Chaitanya K Joshi, Thomas Laurent, Yoshua Bengio,  and Xavier Bresson, “Benchmarking graph neural networks,” arXiv preprint arXiv:2003.00982  (2020).
  89. Christopher Morris, Martin Ritzert, Matthias Fey, William L. Hamilton, Jan Eric Lenssen, Gaurav Rattan,  and Martin Grohe, “Weisfeiler and leman go neural: Higher-order graph neural networks,”  (2021), arXiv:1810.02244 [cs.LG] .
  90. Matthias Fey and Jan Eric Lenssen, “Fast graph representation learning with pytorch geometric,”  (2019), arXiv:1903.02428 [cs.LG] .
  91. Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio,  and Yoshua Bengio, “Graph attention networks,” arXiv preprint arXiv:1710.10903  (2017).
  92. Junhyun Lee, Inyeop Lee,  and Jaewoo Kang, “Self-attention graph pooling,”  (2019), arXiv:1904.08082 [cs.LG] .
  93. Boris Knyazev, Graham W. Taylor,  and Mohamed R. Amer, “Understanding attention and generalization in graph neural networks,”  (2019), arXiv:1905.02850 [cs.LG] .
  94. Hongyang Gao and Shuiwang Ji, “Graph u-nets,”  (2019), arXiv:1905.05178 [cs.LG] .
  95. Cătălina Cangea, Petar Veličković, Nikola Jovanović, Thomas Kipf,  and Pietro Liò, “Towards sparse hierarchical graph classifiers,”  (2018), arXiv:1811.01287 [stat.ML] .
  96. David Kingsley Tuckett, Tailoring surface codes: Improvements in quantum error correction with biased noise, Ph.D. thesis, University of Sydney (2020), (qecsim: https://github.com/qecsim/qecsim).
  97. David K. Tuckett, Stephen D. Bartlett,  and Steven T. Flammia, “Ultrahigh error threshold for surface codes with biased noise,” Physical Review Letters 120 (2018), 10.1103/physrevlett.120.050505.
  98. J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia,  and Benjamin J. Brown, “The XZZX surface code,” Nature Communications 12, 2172 (2021).
  99. Arpit Dua, Aleksander Kubica, Liang Jiang, Steven T. Flammia,  and Michael J. Gullans, “Clifford-deformed surface codes,”  (2022).
  100. Konstantin Tiurev, Peter-Jan H. S. Derks, Joschka Roffe, Jens Eisert,  and Jan-Michael Reiner, “Correcting non-independent and non-identically distributed errors with surface codes,”  (2022).
  101. Eric Huang, Arthur Pesah, Christopher T. Chubb, Michael Vasmer,  and Arpit Dua, “Tailoring three-dimensional topological codes for biased noise,”  (2022).
  102. H. Bombin and M. A. Martin-Delgado, “Topological quantum distillation,” Physical Review Letters 97 (2006), 10.1103/physrevlett.97.180501.
  103. Héctor Bombín, ‘‘Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes,” New Journal of Physics 17, 083002 (2015).
  104. James R Wootton, “A family of stabilizer codes for D⁢(Z2)𝐷subscript𝑍2D(Z_{2})italic_D ( italic_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) anyons and majorana modes,” Journal of Physics A: Mathematical and Theoretical 48, 215302 (2015).
  105. James R. Wootton, “Hexagonal matching codes with 2-body measurements,”  (2021), arXiv:2109.13308 .
  106. Basudha Srivastava, Anton Frisk Kockum,  and Mats Granath, “The XYZ22{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT hexagonal stabilizer code,” Quantum 6, 698 (2022).
  107. Bence Hetényi and James R. Wootton, “Tailoring quantum error correction to spin qubits,”  (2023).
  108. Jeongwan Haah and Matthew B. Hastings, “Boundaries for the Honeycomb Code,”  (2021), arXiv:2110.09545 .
  109. 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,”  (2022).
  110. 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 (2022), 10.1103/physreva.106.062428.
  111. Konstantin Tiurev, Peter-Jan H. S. Derks, Joschka Roffe, Jens Eisert,  and Jan-Michael Reiner, “Correcting non-independent and non-identically distributed errors with surface codes,”  (2023), arXiv:2208.02191 [quant-ph] .
  112. https://github.com/LangeMoritz/GNN_decoder .
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