Quantum error correction for the toric code using deep reinforcement learning

Philip Andreasson; Joel Johansson; Simon Liljestrand; Mats Granath

doi:10.22331/q-2019-09-02-183

Quantum error correction for the toric code using deep reinforcement learning

Philip Andreasson, Joel Johansson, Simon Liljestrand, and Mats Granath

Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden

Published:	2019-09-02, volume 3, page 183
Eprint:	arXiv:1811.12338v3
Doi:	https://doi.org/10.22331/q-2019-09-02-183
Citation:	Quantum 3, 183 (2019).

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

Abstract

We implement a quantum error correction algorithm for bit-flip errors on the topological toric code using deep reinforcement learning. An action-value Q-function encodes the discounted value of moving a defect to a neighboring site on the square grid (the action) depending on the full set of defects on the torus (the syndrome or state). The Q-function is represented by a deep convolutional neural network. Using the translational invariance on the torus allows for viewing each defect from a central perspective which significantly simplifies the state space representation independently of the number of defect pairs. The training is done using experience replay, where data from the algorithm being played out is stored and used for mini-batch upgrade of the Q-network. We find performance which is close to, and for small error rates asymptotically equivalent to, that achieved by the Minimum Weight Perfect Matching algorithm for code distances up to $d=7$. Our results show that it is possible for a self-trained agent without supervision or support algorithms to find a decoding scheme that performs on par with hand-made algorithms, opening up for future machine engineered decoders for more general error models and error correcting codes.

Featured image: Syndrome diagnosing the state of a quantum code for an error protected quantum memory. Arrows indicate the reinforcement learning action-value for a move that corresponds to a single qubit bit-flip operation.

Popular summary

Quantum computers are much more susceptible to noise than present day classical computers. To construct a universal quantum computer it will be necessary to incorporate an auxiliary system for error correction, otherwise errors would quickly accumulate and ruin the calculation. Errors of the quantum bits, or qubits, are the quantum analogs of bit flip errors that also occur in a classical computer. However, in contrast to the classical bits, it is not possible to get an exact diagnosis of qubit errors without destroying the stored quantum information. Instead the error correction has to rely on partial information known as the syndrome and based on this suggest the best way to correct errors. Because of the incomplete information this is a very challenging problem requiring sophisticated algorithms known as error decoders.
In this paper we develop an error decoder based on artificial intelligence. We use deep reinforcement learning, which is the same framework that has recently achieved super-human performance in playing computer and board games. By exploration, experience is gathered and used to train an artificial neural network that can suggest the best error correction to perform for any given syndrome. Our results show that it is possible for a self-trained agent without supervision or support algorithms to find a decoding scheme that performs on par with hand-made algorithms, opening up for future machine engineered decoders for more general types of noise and error correcting codes.

► BibTeX data

@article{Andreasson2019quantumerror,
  doi = {10.22331/q-2019-09-02-183},
  url = {https://doi.org/10.22331/q-2019-09-02-183},
  title = {Quantum error correction for the toric code using deep reinforcement learning},
  author = {Andreasson, Philip and Johansson, Joel and Liljestrand, Simon and Granath, Mats},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {183},
  month = sep,
  year = {2019}
}

► References

[1] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems 25, pages 1097–1105, 2012.

[2] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521 (7553): 436, 2015. 10.1038/nature14539.
https://doi.org/10.1038/nature14539

[3] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, 2016. http://www.deeplearningbook.org.
http://www.deeplearningbook.org

[4] Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.

[5] Gerald Tesauro. Temporal difference learning and td-gammon. Communications of the ACM, 38 (3): 58–68, 1995. URL https://link.galegroup.com/apps/doc/A16764437/AONE?u=googlescholar&sid=AONE&xid=f888cd62.
https://link.galegroup.com/apps/doc/A16764437/AONE?u=googlescholar&sid=AONE&xid=f888cd62

[6] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller. Playing atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602, 2013. URL https://arxiv.org/abs/1312.5602.
arXiv:1312.5602

[7] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518 (7540): 529, 2015. 10.1038/nature14236.
https://doi.org/10.1038/nature14236

[8] David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550 (7676): 354, 2017. 10.1038/nature24270.
https://doi.org/10.1038/nature24270

[9] Louis-François Arsenault, Alejandro Lopez-Bezanilla, O Anatole von Lilienfeld, and Andrew J Millis. Machine learning for many-body physics: the case of the anderson impurity model. Physical Review B, 90 (15): 155136, 2014. 10.1103/PhysRevB.90.155136.
https://doi.org/10.1103/PhysRevB.90.155136

[10] Evert PL Van Nieuwenburg, Ye-Hua Liu, and Sebastian D Huber. Learning phase transitions by confusion. Nature Physics, 13 (5): 435, 2017. 10.1038/nphys4037.
https://doi.org/10.1038/nphys4037

[11] Juan Carrasquilla and Roger G Melko. Machine learning phases of matter. Nature Physics, 13 (5): 431, 2017. 10.1038/nphys4035.
https://doi.org/10.1038/nphys4035

[12] Giuseppe Carleo and Matthias Troyer. Solving the quantum many-body problem with artificial neural networks. Science, 355 (6325): 602–606, 2017. 10.1126/science.aag2302.
https://doi.org/10.1126/science.aag2302

[13] Xun Gao and Lu-Ming Duan. Efficient representation of quantum many-body states with deep neural networks. Nature communications, 8 (1): 662, 2017. 10.1038/s41467-017-00705-2.
https://doi.org/10.1038/s41467-017-00705-2

[14] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52: R2493–R2496, Oct 1995. 10.1103/PhysRevA.52.R2493.
https://doi.org/10.1103/PhysRevA.52.R2493

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

[16] Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information, 2002.

[17] Barbara M Terhal. Quantum error correction for quantum memories. Reviews of Modern Physics, 87 (2): 307, 2015. 10.1103/RevModPhys.87.307.
https://doi.org/10.1103/RevModPhys.87.307

[18] Alexey A Melnikov, Hendrik Poulsen Nautrup, Mario Krenn, Vedran Dunjko, Markus Tiersch, Anton Zeilinger, and Hans J Briegel. Active learning machine learns to create new quantum experiments. Proceedings of the National Academy of Sciences, 115 (6): 1221–1226, 2018. 10.1073/pnas.1714936115.
https://doi.org/10.1073/pnas.1714936115

[19] Thomas Fösel, Petru Tighineanu, Talitha Weiss, and Florian Marquardt. Reinforcement learning with neural networks for quantum feedback. Phys. Rev. X, 8: 031084, Sep 2018. 10.1103/PhysRevX.8.031084.
https://doi.org/10.1103/PhysRevX.8.031084

[20] Marin Bukov, Alexandre G. R. Day, Dries Sels, Phillip Weinberg, Anatoli Polkovnikov, and Pankaj Mehta. Reinforcement learning in different phases of quantum control. Phys. Rev. X, 8: 031086, Sep 2018. 10.1103/PhysRevX.8.031086.
https://doi.org/10.1103/PhysRevX.8.031086

[21] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning. Nature, 549 (7671): 195, 2017. 10.1038/nature23474.
https://doi.org/10.1038/nature23474

[22] A Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2–30, 2003. 10.1016/S0003-4916(02)00018-0.
https://doi.org/10.1016/S0003-4916(02)00018-0

[23] 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.
https://doi.org/10.1063/1.1499754

[24] Robert Raussendorf, Jim Harrington, and Kovid Goyal. Topological fault-tolerance in cluster state quantum computation. New Journal of Physics, 9 (6): 199, 2007. 10.1088/1367-2630/9/6/199.
https://doi.org/10.1088/1367-2630/9/6/199

[25] 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.
https://doi.org/10.1103/PhysRevA.86.032324

[26] Julian Kelly, Rami Barends, Austin G Fowler, Anthony Megrant, Evan Jeffrey, Theodore C White, Daniel Sank, Josh Y Mutus, Brooks Campbell, Yu Chen, et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature, 519 (7541): 66, 2015. 10.1038/nature14270.
https://doi.org/10.1038/nature14270

[27] Jack Edmonds. Paths, trees, and flowers. Canadian Journal of mathematics, 17 (3): 449–467, 1965. 10.4153/CJM-1965-045-4.
https://doi.org/10.4153/CJM-1965-045-4

[28] 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 (1&2): 0145–0158, 2015. URL http://dl.acm.org/citation.cfm?id=2685188.2685197.
http://dl.acm.org/citation.cfm?id=2685188.2685197

[29] Sergey Bravyi, Martin Suchara, and Alexander Vargo. Efficient algorithms for maximum likelihood decoding in the surface code. Phys. Rev. A, 90: 032326, Sep 2014. 10.1103/PhysRevA.90.032326.
https://doi.org/10.1103/PhysRevA.90.032326

[30] Ryan Sweke, Markus S Kesselring, Evert PL van Nieuwenburg, and Jens Eisert. Reinforcement learning decoders for fault-tolerant quantum computation. arXiv preprint arXiv:1810.07207, 2018. URL https://arxiv.org/abs/1810.07207.
arXiv:1810.07207

[31] Guillaume Duclos-Cianci and David Poulin. Fast decoders for topological quantum codes. Physical review letters, 104 (5): 050504, 2010. 10.1103/PhysRevLett.104.050504.
https://doi.org/10.1103/PhysRevLett.104.050504

[32] Guillaume Duclos-Cianci and David Poulin. Fault-tolerant renormalization group decoder for abelian topological codes. Quantum Info. Comput., 14 (9&10): 721–740, July 2014. ISSN 1533-7146. URL http://dl.acm.org/citation.cfm?id=2638670.2638671.
http://dl.acm.org/citation.cfm?id=2638670.2638671

[33] Michael Herold, Earl T Campbell, Jens Eisert, and Michael J Kastoryano. Cellular-automaton decoders for topological quantum memories. npj Quantum Information, 1: 15010, 2015. 10.1038/npjqi.2015.10.
https://doi.org/10.1038/npjqi.2015.10

[34] Aleksander Kubica and John Preskill. Cellular-automaton decoders with provable thresholds for topological codes. arXiv preprint arXiv:1809.10145, 2018. URL https://arxiv.org/abs/1809.10145. 10.1103/PhysRevLett.123.020501.
https://doi.org/10.1103/PhysRevLett.123.020501
arXiv:1809.10145

[35] Giacomo Torlai and Roger G. Melko. Neural decoder for topological codes. Phys. Rev. Lett., 119: 030501, Jul 2017. 10.1103/PhysRevLett.119.030501.
https://doi.org/10.1103/PhysRevLett.119.030501

[36] Stefan Krastanov and Liang Jiang. Deep neural network probabilistic decoder for stabilizer codes. Scientific reports, 7 (1): 11003, 2017. 10.1038/s41598-017-11266-1.
https://doi.org/10.1038/s41598-017-11266-1

[37] Savvas Varsamopoulos, Ben Criger, and Koen Bertels. Decoding small surface codes with feedforward neural networks. Quantum Science and Technology, 3 (1): 015004, 2017. 10.1088/2058-9565/aa955a.
https://doi.org/10.1088/2058-9565/aa955a

[38] 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. 10.22331/q-2018-01-29-48.
https://doi.org/10.22331/q-2018-01-29-48

[39] Nikolas P Breuckmann and Xiaotong Ni. Scalable neural network decoders for higher dimensional quantum codes. Quantum, 2: 68, 2018. 10.22331/q-2018-05-24-68.
https://doi.org/10.22331/q-2018-05-24-68

[40] Christopher Chamberland and Pooya Ronagh. Deep neural decoders for near term fault-tolerant experiments. Quantum Sci. Technol., 3: 044002, 2018. 10.1088/2058-9565/aad1f7.
https://doi.org/10.1088/2058-9565/aad1f7

[41] Nishad Maskara, Aleksander Kubica, and Tomas Jochym-O'Connor. Advantages of versatile neural-network decoding for topological codes. Phys. Rev. A, 99: 052351, May 2019. 10.1103/PhysRevA.99.052351.
https://doi.org/10.1103/PhysRevA.99.052351

[42] Xiaotong Ni. Neural network decoders for large-distance 2d toric codes. arXiv preprint arXiv:1809.06640, 2018. URL https://arxiv.org/abs/1809.06640.
arXiv:1809.06640

[43] Ye-Hua Liu and David Poulin. Neural belief-propagation decoders for quantum error-correcting codes. Phys. Rev. Lett., 122: 200501, May 2019. 10.1103/PhysRevLett.122.200501.
https://doi.org/10.1103/PhysRevLett.122.200501

[44] Dan Browne. Topological codes and computation a lecture course given at the university of innsbruck. 2014. URL http://bit.do/topological.
http://bit.do/topological

[45] David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia. Ultrahigh error threshold for surface codes with biased noise. Phys. Rev. Lett., 120: 050505, Jan 2018. 10.1103/PhysRevLett.120.050505.
https://doi.org/10.1103/PhysRevLett.120.050505

[46] Vladimir Kolmogorov. Blossom v: a new implementation of a minimum cost perfect matching algorithm. Mathematical Programming Computation, 1 (1): 43–67, 2009. 10.1007/s12532-009-0002-8.
https://doi.org/10.1007/s12532-009-0002-8

[47] Austin G Fowler. Optimal complexity correction of correlated errors in the surface code. arXiv preprint arXiv:1310.0863, 2013. URL https://arxiv.org/abs/1310.0863.
arXiv:1310.0863

[48] Mattias Eliasson, David Fitzek, and Mats Granath. In preperation, 2019.

[49] Dan Horgan, John Quan, David Budden, Gabriel Barth-Maron, Matteo Hessel, Hado Van Hasselt, and David Silver. Distributed prioritized experience replay. arXiv preprint arXiv:1803.00933, 2018. URL https://arxiv.org/abs/1803.00933.
arXiv:1803.00933

Cited by

[1] Hossein Dehghani, Ali Lavasani, Mohammad Hafezi, and Michael J. Gullans, "Neural-network decoders for measurement induced phase transitions", Nature Communications 14 1, 2918 (2023).

[2] Meriam Gay Bautista, Zhi Jackie Yao, Anastasiia Butko, Mariam Kiran, and Mekena Metcalf, 2021 IEEE Computer Society Annual Symposium on VLSI (ISVLSI) 462 (2021) ISBN:978-1-6654-3946-6.

[3] Spiro Gicev, Lloyd C. L. Hollenberg, and Muhammad Usman, "A scalable and fast artificial neural network syndrome decoder for surface codes", Quantum 7, 1058 (2023).

[4] Agnes Valenti, Evert van Nieuwenburg, Sebastian Huber, and Eliska Greplova, "Hamiltonian learning for quantum error correction", Physical Review Research 1 3, 033092 (2019).

[5] Hao-Wen Wang , Qian Cao , Yun-Jia Xue , Li Ding , Han-Yang Liu , Yu-Min Dong , and Hong-Yang Ma , "Determining quantum topological semion code decoder performance and error correction effectiveness with reinforcement learning", Frontiers in Physics 10, 981225 (2022).

[6] Chenfeng Cao, Chao Zhang, Zipeng Wu, Markus Grassl, and Bei Zeng, "Quantum variational learning for quantum error-correcting codes", Quantum 6, 828 (2022).

[7] Juan Carrasquilla and Giacomo Torlai, "How To Use Neural Networks To Investigate Quantum Many-Body Physics", PRX Quantum 2 4, 040201 (2021).

[8] 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", Physical Review A 105 4, 042616 (2022).

[9] Xiao-Ming Zhang, Zezhu Wei, Raza Asad, Xu-Chen Yang, and Xin Wang, "When does reinforcement learning stand out in quantum control? A comparative study on state preparation", npj Quantum Information 5 1, 85 (2019).

[10] Aoqing Li, Fan Li, Qidi Gan, and Hongyang Ma, "Convolutional-Neural-Network-Based Hexagonal Quantum Error Correction Decoder", Applied Sciences 13 17, 9689 (2023).

[11] Hugo Théveniaut and Everard van Nieuwenburg, "A NEAT quantum error decoder", SciPost Physics 11 1, 005 (2021).

[12] Oleksandr Balabanov and Mats Granath, "Unsupervised learning using topological data augmentation", Physical Review Research 2 1, 013354 (2020).

[13] Xianchao Zhu and Xiaokai Hou, "Quantum architecture search via truly proximal policy optimization", Scientific Reports 13 1, 5157 (2023).

[14] Juan Carrasquilla, "Machine learning for quantum matter", Advances in Physics: X 5 1, 1797528 (2020).

[15] 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 2, 023230 (2020).

[16] Tomi Ohtsuki and Tomohiro Mano, "Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions", Journal of the Physical Society of Japan 89 2, 022001 (2020).

[17] Sofiene Jerbi, Lea M. Trenkwalder, Hendrik Poulsen Nautrup, Hans J. Briegel, and Vedran Dunjko, "Quantum Enhancements for Deep Reinforcement Learning in Large Spaces", PRX Quantum 2 1, 010328 (2021).

[18] Samuel Yen-Chi Chen, Chao-Han Huck Yang, Jun Qi, Pin-Yu Chen, Xiaoli Ma, and Hsi-Sheng Goan, "Variational Quantum Circuits for Deep Reinforcement Learning", IEEE Access 8, 141007 (2020).

[19] V. V. Sivak, A. Eickbusch, H. Liu, B. Royer, I. Tsioutsios, and M. H. Devoret, "Model-Free Quantum Control with Reinforcement Learning", Physical Review X 12 1, 011059 (2022).

[20] Haowen Wang, Zhaoyang Song, Yinuo Wang, Yanbing Tian, and Hongyang Ma, "Target-generating quantum error correction coding scheme based on generative confrontation network", Quantum Information Processing 21 8, 280 (2022).

[21] Ying-Jie 英杰 Qu 曲, Zhao 钊 Chen 陈, Wei-Jie 伟杰 Wang 王, and Hong-Yang 鸿洋 Ma 马, "Approximate error correction scheme for three-dimensional surface codes based reinforcement learning", Chinese Physics B 32 10, 100307 (2023).

[22] Hiroki Saito, "Creation and Manipulation of Quantized Vortices in Bose–Einstein Condensates Using Reinforcement Learning", Journal of the Physical Society of Japan 89 7, 074006 (2020).

[23] Ran-Yi-Liu Chen, Ben-Chi Zhao, Zhi-Xin Song, Xuan-Qiang Zhao, Kun Wang, and Xin Wang, "Hybrid quantum-classical algorithms: Foundation, design and applications", Acta Physica Sinica 70 21, 210302 (2021).

[24] Colin Bellinger, Rory Coles, Mark Crowley, and Isaac Tamblyn, Lecture Notes in Computer Science 12109, 55 (2020) ISBN:978-3-030-47357-0.

[25] Chris Beeler, Uladzimir Yahorau, Rory Coles, Kyle Mills, Stephen Whitelam, and Isaac Tamblyn, "Optimizing thermodynamic trajectories using evolutionary and gradient-based reinforcement learning", Physical Review E 104 6, 064128 (2021).

[26] Andrey Zhukov and Walter Pogosov, "Quantum error reduction with deep neural network applied at the post-processing stage", Quantum Information Processing 21 3, 93 (2022).

[27] Shoaib Balouch, Muhammad Abrar, Hafiz Abdul Muqeet, Muhammad Shahzad, Harun Jamil, Monia Hamdi, Abdul Sattar Malik, and Habib Hamam, "Optimal Scheduling of Demand Side Load Management of Smart Grid Considering Energy Efficiency", Frontiers in Energy Research 10, 861571 (2022).

[28] S. Varona and M. A. Martin-Delgado, "Determination of the semion code threshold using neural decoders", Physical Review A 102 3, 032411 (2020).

[29] Laia Domingo Colomer, Michalis Skotiniotis, and Ramon Muñoz-Tapia, "Reinforcement learning for optimal error correction of toric codes", Physics Letters A 384 17, 126353 (2020).

[30] Chae-Yeun Park and Michael J. Kastoryano, "Geometry of learning neural quantum states", Physical Review Research 2 2, 023232 (2020).

[31] Oleksandr Balabanov and Mats Granath, "Unsupervised interpretable learning of topological indices invariant under permutations of atomic bands", Machine Learning: Science and Technology 2 2, 025008 (2021).

[32] Friederike Metz and Marin Bukov, "Self-correcting quantum many-body control using reinforcement learning with tensor networks", Nature Machine Intelligence 5 7, 780 (2023).

[33] Ryan Sweke, Markus S Kesselring, Evert P L van Nieuwenburg, and Jens Eisert, "Reinforcement learning decoders for fault-tolerant quantum computation", Machine Learning: Science and Technology 2 2, 025005 (2021).

[34] Li Ding, Haowen Wang, Yinuo Wang, Shumei Wang, and YuBo Sheng, "Based on Quantum Topological Stabilizer Color Code Morphism Neural Network Decoder", Quantum Engineering 2022, 1 (2022).

[35] 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).

[36] Lorenzo Moro, Matteo G. A. Paris, Marcello Restelli, and Enrico Prati, "Quantum compiling by deep reinforcement learning", Communications Physics 4 1, 178 (2021).

[37] F Battistel, C Chamberland, K Johar, R W J Overwater, F Sebastiano, L Skoric, Y Ueno, and M Usman, "Real-time decoding for fault-tolerant quantum computing: progress, challenges and outlook", Nano Futures 7 3, 032003 (2023).

[38] Jia-Hao Cao, Feng Chen, Qi Liu, Tian-Wei Mao, Wen-Xin Xu, Ling-Na Wu, and Li You, "Detection of Entangled States Supported by Reinforcement Learning", Physical Review Letters 131 7, 073201 (2023).

[39] Jelena Mackeprang, Durga B. Rao Dasari, and Jörg Wrachtrup, "A reinforcement learning approach for quantum state engineering", Quantum Machine Intelligence 2 1, 5 (2020).

[40] Zaigham Mushtaq, Muhammad Farhan Ramzan, Sikandar Ali, Samad Baseer, Ali Samad, Mujtaba Husnain, and Ahmed Farouk, "Voting Classification-Based Diabetes Mellitus Prediction Using Hypertuned Machine-Learning Techniques", Mobile Information Systems 2022, 1 (2022).

[41] Sumaira Ahmed, Salahuddin Shaikh, Farwa Ikram, Muhammad Fayaz, Hathal Salamah Alwageed, Faheem Khan, Fawwad Hassan Jaskani, and Rajesh Kaluri, "Prediction of Cardiovascular Disease on Self-Augmented Datasets of Heart Patients Using Multiple Machine Learning Models", Journal of Sensors 2022, 1 (2022).

[42] Poulami Das, Christopher A. Pattison, Srilatha Manne, Douglas M. Carmean, Krysta M. Svore, Moinuddin Qureshi, and Nicolas Delfosse, 2022 IEEE International Symposium on High-Performance Computer Architecture (HPCA) 259 (2022) ISBN:978-1-6654-2027-3.

[43] Chae-Yeun Park and Michael J. Kastoryano, "Expressive power of complex-valued restricted Boltzmann machines for solving nonstoquastic Hamiltonians", Physical Review B 106 13, 134437 (2022).

[44] Lucas Lamata, "Quantum Reinforcement Learning with Quantum Photonics", Photonics 8 2, 33 (2021).

[45] Kai Meinerz, Chae-Yeun Park, and Simon Trebst, "Scalable Neural Decoder for Topological Surface Codes", Physical Review Letters 128 8, 080505 (2022).

[46] Christopher Chamberland, Luis Goncalves, Prasahnt Sivarajah, Eric Peterson, and Sebastian Grimberg, "Techniques for combining fast local decoders with global decoders under circuit-level noise", Quantum Science and Technology 8 4, 045011 (2023).

[47] Jonas Schuff, Lukas J Fiderer, and Daniel Braun, "Improving the dynamics of quantum sensors with reinforcement learning", New Journal of Physics 22 3, 035001 (2020).

[48] Hendrik Poulsen Nautrup, Nicolas Delfosse, Vedran Dunjko, Hans J. Briegel, and Nicolai Friis, "Optimizing Quantum Error Correction Codes with Reinforcement Learning", arXiv:1812.08451, (2018).

[49] Jun-Jie Chen and Ming Xue, "Manipulation of Spin Dynamics by Deep Reinforcement Learning Agent", arXiv:1901.08748, (2019).

[50] Chaitanya Chinni, Abhishek Kulkarni, Dheeraj M. Pai, Kaushik Mitra, and Pradeep Kiran Sarvepalli, "Neural Decoder for Topological Codes using Pseudo-Inverse of Parity Check Matrix", arXiv:1901.07535, (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2023-11-28 21:01:39) and SAO/NASA ADS (last updated successfully 2023-11-28 21:01:40). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.