Scalable Neural Network Decoders for Higher Dimensional Quantum Codes

Nikolas P. Breuckmann; Xiaotong Ni

doi:10.22331/q-2018-05-24-68

Scalable Neural Network Decoders for Higher Dimensional Quantum Codes

Nikolas P. Breuckmann¹ and Xiaotong Ni^1,2

¹Institute for Quantum Information, RWTH Aachen University, Germany
²Max Planck Institute of Quantum Optics, Germany

Published:	2018-05-24, volume 2, page 68
Eprint:	arXiv:1710.09489v3
Doi:	https://doi.org/10.22331/q-2018-05-24-68
Citation:	Quantum 2, 68 (2018).

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

Abstract

Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that they can be evaluated by dedicated hardware which is very fast and consumes little power. Machine learning has been previously applied to decode the surface code. However, these approaches are not scalable as the training has to be redone for every system size which becomes increasingly difficult. In this work the existence of local decoders for higher dimensional codes leads us to use a low-depth convolutional neural network to locally assign a likelihood of error on each qubit. For noiseless syndrome measurements, numerical simulations show that the decoder has a threshold of around 7.1% when applied to the 4D toric code. When the syndrome measurements are noisy, the decoder performs better for larger code sizes when the error probability is low. We also give theoretical and numerical analysis to show how a convolutional neural network is different from the 1-nearest neighbor algorithm, which is a baseline machine learning method.

► BibTeX data

@article{Breuckmann2018scalableneural,
  doi = {10.22331/q-2018-05-24-68},
  url = {https://doi.org/10.22331/q-2018-05-24-68},
  title = {Scalable {N}eural {N}etwork {D}ecoders for {H}igher {D}imensional {Q}uantum {C}odes},
  author = {Breuckmann, Nikolas P. and Ni, Xiaotong},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {68},
  month = may,
  year = {2018}
}

► References

[1] Charlene Sonja Ahn. Extending quantum error correction: new continuous measurement protocols and improved fault-tolerant overhead. PhD thesis, California Institute of Technology, 2004.

[2] Gaku Arakawa, Ikuo Ichinose, Tetsuo Matsui, and Koujin Takeda. Self-duality and phase structure of the 4d random-plaquette z2 gauge model. Nuclear Physics B, 709 (1): 296–306, 2005. 10.1016/j.nuclphysb.2004.12.024.
https://doi.org/10.1016/j.nuclphysb.2004.12.024

[3] Paul Baireuther, Thomas E. O'Brien, Brian Tarasinski, and Carlo W. J. Beenakker. Machine-learning-assisted correction of correlated qubit errors in a topological code. Quantum, 2: 48, jan 2018. 10.22331/q-2018-01-29-48.
https://doi.org/10.22331/q-2018-01-29-48

[4] Yoshua Bengio, Yann LeCun, et al. Scaling learning algorithms towards ai. Large-scale kernel machines, 34 (5): 1–41, 2007.

[5] Héctor Bombín. Single-shot fault-tolerant quantum error correction. Physical Review X, 5 (3): 031043, 2015. 10.1103/PhysRevX.5.031043.
https://doi.org/10.1103/PhysRevX.5.031043

[6] Nikolas P. Breuckmann. Homological quantum codes beyond the toric code. PhD thesis, RWTH Aachen University, 2017. URL https://doi.org/10.18154/rwth-2018-01100.
https://doi.org/10.18154/rwth-2018-01100

[7] Nikolas P Breuckmann, Kasper Duivenvoorden, Dominik Michels, and Barbara M Terhal. Local decoders for the 2d and 4d toric code. Quantum Information and Computation, 17 (3 and 4): 0181–0208, 2017. 10.26421/QIC17.3-4.
https://doi.org/10.26421/QIC17.3-4

[8] Christopher Clark and Amos Storkey. Training deep convolutional neural networks to play go. In International Conference on Machine Learning, pages 1766–1774, 2015.

[9] Joshua Combes, Christopher Ferrie, Chris Cesare, Markus Tiersch, GJ Milburn, Hans J Briegel, and Carlton M Caves. In-situ characterization of quantum devices with error correction. 2014. URL https://arxiv.org/abs/1405.5656.
arXiv:1405.5656

[10] H.A. David and H.N. Nagaraja. Order Statistics. Wiley Series in Probability and Statistics. Wiley, 2004. ISBN 9780471654018.

[11] 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

[12] 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

[13] Guillaume Duclos-Cianci and David Poulin. Fault-tolerant renormalization group decoder for abelian topological codes. Quantum Information & Computation, 14 (9-10): 721–740, 2014.

[14] Kasper Duivenvoorden, Nikolas P Breuckmann, and Barbara M Terhal. Renormalization group decoder for a four-dimensional toric code. arXiv preprint arXiv:1708.09286, 2017.
arXiv:1708.09286

[15] Brendan J Frey and David JC MacKay. A revolution: Belief propagation in graphs with cycles. Advances in neural information processing systems, pages 479–485, 1998.

[16] Gabriel Goh. Why momentum really works. Distill, 2017. 10.23915/distill.00006.
https://doi.org/10.23915/distill.00006

[17] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, 2016.

[18] Matthew B Hastings. Decoding in hyperbolic spaces: Ldpc codes with linear rate and efficient error correction. Quantum Information and Computation, 14, 2014.

[19] Norman P Jouppi, Cliff Young, Nishant Patil, David Patterson, Gaurav Agrawal, Raminder Bajwa, Sarah Bates, Suresh Bhatia, Nan Boden, Al Borchers, et al. In-datacenter performance analysis of a tensor processing unit. 44th International Symposium on Computer Architecture, 2017. 10.1145/3079856.3080246.
https://doi.org/10.1145/3079856.3080246

[20] Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. 3rd International Conference for Learning Representations, San Diego, 2015. URL https://arxiv.org/abs/1412.6980.
arXiv:1412.6980

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

[22] Stephen Marsland. Machine Learning: An Algorithmic Perspective, Second Edition. Chapman & Hall/CRC, 2nd edition, 2014. ISBN 1466583282, 9781466583283.

[23] Paul A Merolla, John V Arthur, Rodrigo Alvarez-Icaza, Andrew S Cassidy, Jun Sawada, Filipp Akopyan, Bryan L Jackson, Nabil Imam, Chen Guo, Yutaka Nakamura, et al. A million spiking-neuron integrated circuit with a scalable communication network and interface. Science, 345 (6197): 668–673, 2014. 10.1126/science.1254642.
https://doi.org/10.1126/science.1254642

[24] Janardan Misra and Indranil Saha. Artificial neural networks in hardware: A survey of two decades of progress. Neurocomputing, 74 (1–3): 239 – 255, 2010. ISSN 0925-2312. 10.1016/j.neucom.2010.03.021.
https://doi.org/10.1016/j.neucom.2010.03.021

[25] Eliya Nachmani, Yair Be'ery, and David Burshtein. Learning to decode linear codes using deep learning. In 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton). IEEE, sep 2016. 10.1109/allerton.2016.7852251.
https://doi.org/10.1109/allerton.2016.7852251

[26] Michael A. Nielsen. Neural Networks and Deep Learning. Determination Press, 2015.

[27] Genevieve B Orr and Klaus-Robert Müller. Neural networks: tricks of the trade. Springer, 2003.

[28] Fernando Pastawski. Quantum memory: design and applications. PhD thesis, LMU Munich, 2012. URL https://edoc.ub.uni-muenchen.de/14703/.
https://edoc.ub.uni-muenchen.de/14703/

[29] David Poulin and Yeojin Chung. On the iterative decoding of sparse quantum codes. Quantum Information and Computation, 8 (10): 0987–1000, 2008.

[30] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. Nature, 529 (7587): 484–489, 2016. 10.1038/nature16961.
https://doi.org/10.1038/nature16961

[31] David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, Yutian Chen, Timothy Lillicrap, Fan Hui, Laurent Sifre, George van den Driessche, Thore Graepel, and Demis Hassabis. Mastering the game of go without human knowledge. Nature, 550 (7676): 354–359, Oct 2017. ISSN 0028-0836. 10.1038/nature24270.
https://doi.org/10.1038/nature24270

[32] John M Sullivan. A crystalline approximation theorem for hypersurfaces. PhD thesis, Princeton University, 1990.

[33] Koujin Takeda and Hidetoshi Nishimori. Self-dual random-plaquette gauge model and the quantum toric code. Nuclear Physics B, 686 (3): 377 – 396, 2004. ISSN 0550-3213. 10.1016/j.nuclphysb.2004.03.006.
https://doi.org/10.1016/j.nuclphysb.2004.03.006

[34] David Barrie Thomas, Lee Howes, and Wayne Luk. A comparison of cpus, gpus, fpgas, and massively parallel processor arrays for random number generation. In Proceedings of the ACM/SIGDA International Symposium on Field Programmable Gate Arrays, FPGA '09, pages 63–72, New York, NY, USA, 2009. ACM. ISBN 978-1-60558-410-2. 10.1145/1508128.1508139.
https://doi.org/10.1145/1508128.1508139

[35] 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.
https://doi.org/10.1103/PhysRevA.90.062320

[36] Giacomo Torlai and Roger G Melko. A neural decoder for topological codes. Physical Review Letters, 119 (3): 030501, 2017. 10.1103/PhysRevLett.119.030501.
https://doi.org/10.1103/PhysRevLett.119.030501

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

[38] Chenyang Wang, Jim Harrington, and John Preskill. Confinement-higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory. Annals of Physics, 303 (1): 31–58, 2003. 10.1016/S0003-4916(02)00019-2.
https://doi.org/10.1016/S0003-4916(02)00019-2

[39] Jonathan S Yedidia, William T Freeman, and Yair Weiss. Understanding belief propagation and its generalizations. Exploring artificial intelligence in the new millennium, 8: 236–239, 2003.

[40] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. 5th International Conference on Learning Representations, 2016.

Cited by

[1] 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.

[2] 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).

[3] 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 1, 013003 (2019).

[4] 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).

[5] Savvas Varsamopoulos, Koen Bertels, and Carmen Garcia Almudever, "Comparing Neural Network Based Decoders for the Surface Code", IEEE Transactions on Computers 69 2, 300 (2020).

[6] Benjamin J. Brown and Dominic J. Williamson, "Parallelized quantum error correction with fracton topological codes", Physical Review Research 2 1, 013303 (2020).

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

[8] Philip Andreasson, Joel Johansson, Simon Liljestrand, and Mats Granath, "Quantum error correction for the toric code using deep reinforcement learning", Quantum 3, 183 (2019).

[9] Chaitanya Chinni, Abhishek Kulkami, Dheeraj MPai, Kaushik Mitra, and Pradeep K Sarvepalli, 2019 IEEE Information Theory Workshop (ITW) 1 (2019) ISBN:978-1-5386-6900-6.

[10] Michael Vasmer, Dan E. Browne, and Aleksander Kubica, "Cellular automaton decoders for topological quantum codes with noisy measurements and beyond", Scientific Reports 11 1, 2027 (2021).

[11] G Dauphinais, L Ortiz, S Varona, and M A Martin-Delgado, "Quantum error correction with the semion code", New Journal of Physics 21 5, 053035 (2019).

[12] Yun-Jia Xue, Hao-Wen Wang, Yan-Bing Tian, Yi-Nuo Wang, Yu-Xuan Wang, Shu-Mei Wang, and Liuguo Yin, "Quantum Information Protection Scheme Based on Reinforcement Learning for Periodic Surface Codes", Quantum Engineering 2022, 1 (2022).

[13] V. Saggio, B. E. Asenbeck, A. Hamann, T. Strömberg, P. Schiansky, V. Dunjko, N. Friis, N. C. Harris, M. Hochberg, D. Englund, S. Wölk, H. J. Briegel, and P. Walther, "Experimental quantum speed-up in reinforcement learning agents", Nature 591 7849, 229 (2021).

[14] 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).

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

[16] Aleksander Kubica and Nicolas Delfosse, "Efficient color code decoders in d≥2 dimensions from toric code decoders", Quantum 7, 929 (2023).

[17] T. R. Scruby, D. E. Browne, P. Webster, and M. Vasmer, "Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code", Quantum 6, 721 (2022).

[18] Armanda O. Quintavalle, Michael Vasmer, Joschka Roffe, and Earl T. Campbell, "Single-Shot Error Correction of Three-Dimensional Homological Product Codes", PRX Quantum 2 2, 020340 (2021).

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

[20] Xiaotong Ni, "Neural Network Decoders for Large-Distance 2D Toric Codes", Quantum 4, 310 (2020).

[21] Amarsanaa Davaasuren, Yasunari Suzuki, Keisuke Fujii, and Masato Koashi, "General framework for constructing fast and near-optimal machine-learning-based decoder of the topological stabilizer codes", Physical Review Research 2 3, 033399 (2020).

[22] 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).

[23] K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, "Renormalization Group Decoder for a Four-Dimensional Toric Code", IEEE Transactions on Information Theory 65 4, 2545 (2019).

[24] 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).

[25] Zhaoyi Li, Isaac Kim, and Patrick Hayden, "Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction", Quantum 7, 1089 (2023).

[26] 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).

[27] Leonid P. Pryadko, "On maximum-likelihood decoding with circuit-level errors", Quantum 4, 304 (2020).

[28] Arun B. Aloshious and Pradeep Kiran Sarvepalli, "Decoding Toric Codes on Three Dimensional Simplical Complexes", IEEE Transactions on Information Theory 67 2, 931 (2021).

[29] 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).

[30] Aleksander Kubica and John Preskill, "Cellular-Automaton Decoders with Provable Thresholds for Topological Codes", Physical Review Letters 123 2, 020501 (2019).

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

[32] Ye-Hua Liu and David Poulin, "Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes", Physical Review Letters 122 20, 200501 (2019).

[33] Zhaoyi Li, Issac Kim, and Patrick Hayden, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 773 (2022) ISBN:978-1-6654-9113-6.

[34] Pankaj Mehta, Marin Bukov, Ching-Hao Wang, Alexandre G. R. Day, Clint Richardson, Charles K. Fisher, and David J. Schwab, "A high-bias, low-variance introduction to Machine Learning for physicists", Physics Reports 810, 1 (2019).

[35] 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).

[36] Christopher T. Chubb, "General tensor network decoding of 2D Pauli codes", arXiv:2101.04125, (2021).

[37] 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).

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

The above citations are from Crossref's cited-by service (last updated successfully 2023-11-29 19:30:13) and SAO/NASA ADS (last updated successfully 2023-11-29 19:30:14). 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.