A fault-tolerant quantum computation requires an efficient means to detect and correct errors that accumulate in encoded quantum information. In the context of machine learning, neural networks are a promising new approach to quantum error correction. Here we show that a recurrent neural network can be trained, using only experimentally accessible data, to detect errors in a widely used topological code, the surface code, with a performance above that of the established minimum-weight perfect matching (or blossom) decoder. The performance gain is achieved because the neural network decoder can detect correlations between bit-flip (X) and phase-flip (Z) errors. The machine learning algorithm adapts to the physical system, hence no noise model is needed. The long short-term memory layers of the recurrent neural network maintain their performance over a large number of quantum error correction cycles, making it a practical decoder for forthcoming experimental realizations of the surface code.
 D. A. Lidar, T. A. Brun, editors, Quantum error correction (Cambridge University Press, 2013).
 A. G. Fowler, A. C. Whiteside, and L. C. L. Hollenberg, Towards practical classical processing for the surface code, Phys. Rev. Lett. 108, 180501 (2012).
 A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
 S. Shalev-Shwartz and S. Ben-David, Understanding machine learning: From theory to algorithms (Cambridge University Press, 2014).
 A. G. Fowler, Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $O(1)$ parallel time, Quantum Inf. Comput. 15, 0145 (2015).
 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 J. M. Martinis, State preservation by repetitive error detection in a superconducting quantum circuit, Nature 519, 66 (2015).
 M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, Demonstration of weight-four parity measurements in the surface code architecture, Phys. Rev. Lett. 117, 210505 (2016).
 R. Versluis, S. Poletto, N. Khammassi, B. Tarasinski, N. Haider, D. J. Michalak, A. Bruno, K. Bertels, and L. DiCarlo, Scalable quantum circuit and control for a superconducting surface code, Phys. Rev. Applied 8, 034021 (2017).
 D. Gottesman, Stabilizer codes and quantum error correction (Doctoral dissertation, California Institute of Technology, 1997).
 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). The source code of the quantum simulator can be found at https://github.com/brianzi/quantumsim. The source code of the Surface-17 simulation can be found at https://github.com/obriente/surf17_circuit.
 H. Bombin, and M. A. Martin-Delgado, Optimal resources for topological two-dimensional stabilizer codes: Comparative study, Phys. Rev. A 76, 012305 (2007).
 N. Delfosse and J.-P. Tillich, A decoding algorithm for CSS codes using the X/Z correlations, 2014 IEEE International Symposium on Information Theory, 1071 (2014).
 M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, TensorFlow: Large-scale machine learning on heterogeneous distributed systems, arXiv:1603.04467.
 The source code of the neural network decoder can be found at https://github.com/baireuther/ neural_network_decoder.
 Andrew S. Darmawan and David Poulin, "Linear-time general decoding algorithm for the surface code", Physical Review E 97 5, 051302 (2018).
 Naomi H. Nickerson and Benjamin J. Brown, "Analysing correlated noise on the surface code using adaptive decoding algorithms", Quantum 3, 131 (2019).
 Vedran Dunjko and Hans J Briegel, "Machine learning & artificial intelligence in the quantum domain: a review of recent progress", Reports on Progress in Physics 81 7, 074001 (2018).
 Nikolas P. Breuckmann and Xiaotong Ni, "Scalable Neural Network Decoders for Higher Dimensional Quantum Codes", Quantum 2, 68 (2018).
 Ye-Hua Liu and David Poulin, "Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes", Physical Review Letters 122 20, 200501 (2019).
 Thomas Fösel, Petru Tighineanu, Talitha Weiss, and Florian Marquardt, "Reinforcement Learning with Neural Networks for Quantum Feedback", Physical Review X 8 3, 031084 (2018).
 Kenneth Rudinger, Timothy Proctor, Dylan Langharst, Mohan Sarovar, Kevin Young, and Robin Blume-Kohout, "Probing Context-Dependent Errors in Quantum Processors", Physical Review X 9 2, 021045 (2019).
 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).
 Christopher Chamberland and Pooya Ronagh, "Deep neural decoders for near term fault-tolerant experiments", Quantum Science and Technology 3 4, 044002 (2018).
 S. T. Spitz, B. Tarasinski, C. W. J. Beenakker, and T. E. O'Brien, "Adaptive weight estimator for quantum error correction", arXiv:1712.02360.
 Stefan Krastanov and Liang Jiang, "Deep Neural Network Probabilistic Decoder for Stabilizer Codes", Scientific Reports 7, 11003 (2017).
 Milap Sheth, Sara Zafar Jafarzadeh, and Vlad Gheorghiu, "Neural ensemble decoding for topological quantum error-correcting codes", arXiv:1905.02345.
 David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia, "Ultrahigh Error Threshold for Surface Codes with Biased Noise", Physical Review Letters 120 5, 050505 (2018).
 L. M. K. Vandersypen, H. Bluhm, J. S. Clarke, A. S. Dzurak, R. Ishihara, A. Morello, D. J. Reilly, L. R. Schreiber, and M. Veldhorst, "Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent", npj Quantum Information 3, 34 (2017).
 Hendrik Poulsen Nautrup, Nicolas Delfosse, Vedran Dunjko, Hans J. Briegel, and Nicolai Friis, "Optimizing Quantum Error Correction Codes with Reinforcement Learning", arXiv:1812.08451.
 Christopher T. Chubb and Steven T. Flammia, "Statistical mechanical models for quantum codes with correlated noise", arXiv:1809.10704.
 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", arXiv:1803.08823.
The above citations are from Crossref's cited-by service (last updated 2019-06-18 18:54:48) and SAO/NASA ADS (last updated 2019-06-18 18:54:49). 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.