Quantum Lyapunov control with machine learning

  • S. C. Hou
  • X. X. YiEmail author


Quantum state engineering is a central task in Lyapunov-based quantum control. Given different initial states, better performance may be achieved if the control parameters, such as the Lyapunov function, are individually optimized for each initial state, however, at the expense of computing resources. To tackle this issue, we propose an initial-state-adaptive Lyapunov control strategy with machine learning. Specifically, artificial neural networks are used to learn the relationship between the optimal control parameters and initial states through supervised learning with samples. Two designs are presented where the feedforward neural network and the general regression neural network are used to select control schemes and design Lyapunov functions, respectively. We demonstrate the performance of the designs with a three-level quantum system for an eigenstate control problem. Since the sample generation and the training of neural networks are carried out in advance, the initial-state-adaptive Lyapunov control can be implemented for new initial states without much increase of computational resources.


Lyapunov control Machine learning Neural network Quantum state preparation 



This work is supported by the National Natural Science Foundation of China under Grant No. 11705026, 11534002, 11775048, 61475033, the China Postdoctoral Science Foundation under Grant No. 2017M611293, and the Fundamental Research Funds for the Central Universities under Grant No. 2412017QD003.


  1. 1.
    D’Alessandro, D.: Introduction to Quantum Control and Dynamics. Chapman & Hall, Boca Raton (2007)zbMATHCrossRefGoogle Scholar
  2. 2.
    Wiseman, H.M., Milburn, G.J.: Quantum Measurement and Control. Cambridge University Press, Cambridge (2009)zbMATHCrossRefGoogle Scholar
  3. 3.
    Zhang, J., Liu, Y.-X., Wu, R.-B., Jacobs, K., Nori, F.: Quantum feedback: theory, experiments, and applications. Phys. Rep. 679, 1–60 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Glaser, S.J., Boscain, U., Calarco, T., Koch, C.P., Köckenberger, W., Kosloff, R., Kuprov, I., Luy, B., Schirmer, S., Schulte-Herbrüggen, T., Sugny, D., Wilhelm, F.K.: Training Schrödinger’s cat: quantum optimal control. Eur. Phys. J. D 69, 279 (2015)ADSCrossRefGoogle Scholar
  5. 5.
    Machnes, S., Sander, U., Glaser, S.J., de Fouquières, P., Gruslys, A., Schirmer, S., Schulte-Herbrüggen, T.: Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework. Phys. Rev. A 84, 022305 (2011)ADSCrossRefGoogle Scholar
  6. 6.
    Gough, J.E., Belavkin, V.P.: Quantum control and information processing. Quantum Inf. Process. 12, 1397–1415 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Vettori, P.: On the convergence of a feedback control strategy for multilevel quantum systems. In: Proceedings of the Mathematical Theory of Networks and Systems Conference (2002)Google Scholar
  8. 8.
    Grivopoulos, S., Bamieh, B.: Lyapunov-based control of quantum systems. In: Proceedings of the 42nd IEEE International Conference on Decision and Control, Maui, Hawaii USA, pp. 434–438 (2003)Google Scholar
  9. 9.
    Mirrahimi, M., Rouchon, P., Turinici, G.: Lyapunov control of bilinear Schrödinger equations. Automatica 41, 1987–1994 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kuang, S., Cong, S.: Lyapunov control methods of closed quantum systems. Automatica 44, 98–108 (2008)zbMATHCrossRefGoogle Scholar
  11. 11.
    Wang, X., Schirmer, S.G.: Analysis of Lyapunov method for control of quantum states. IEEE Trans. Autom. Control 55(10), 2259–2270 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hou, S.C., Khan, M.A., Yi, X.X., Dong, D., Petersen, I.R.: Optimal Lyapunov-based quantum control for quantum systems. Phys. Rev. A 86, 022321 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Wang, L.C., Hou, S.C., Yi, X.X., Dong, D., Petersen, I.R.: Optimal Lyapunov quantum control of two-level systems: convergence and extended techniques. Phys. Lett. A 378, 1074 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Zhao, S., Lin, H., Xue, Z.: Switching control of closed quantum systems via the Lyapunov method. Automatica 48(8), 1833–1838 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kuang, S., Dong, D., Petersen, I.R.: Rapid Lyapunov control of finite-dimensional quantum systems. Automatica 81, 164–175 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Silveira, H.B., da Silva, P.S.P., Rouchon, P.: Quantum gate generation for systems with drift in U(n) using Lyapunov–LaSalle techniques. Int. J. Control 89(12), 2466–2481 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Li, W., Li, C., Song, H.: Quantum synchronization in an optomechanical system based on Lyapunov control. Phys. Rev. E 93, 062221 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    Shi, Z.C., Wang, L.C., Yi, X.X.: Preparing entangled states by Lyapunov control. Quantum Inf. Process. 15, 4939–4953 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Shi, Z.C., Zhao, X.L., Yi, X.X.: Preparation of topological modes by Lyapunov control. Sci. Rep. 5, 13777 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    Hou, S.C., Wang, L.C., Yi, X.X.: Realization of quantum gates by Lyapunov control. Phys. Lett. A 378(9), 699–704 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Yi, X.X., Huang, X.L., Wu, C., Oh, C.H.: Driving quantum systems into decoherence-free subspaces by Lyapunov control. Phys. Rev. A 80, 052316 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Amini, H., Somaraju, R.A., Dotsenko, I., Sayrin, C., Mirrahimi, M., Rouchon, P.: Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays. Automatica 49(9), 2683–2692 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Ge, S.S., Vu, T.L., Lee, T.H.: Quantum measurement-based feedback control: a nonsmooth time delay control approach. SIAM J. Control Optim. 50(2), 845–863 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Sayrin, C., Dotsenko, I., Zhou, X., Peaudecerf, B., Rybarczyk, T., Gleyzes, S., Rouchon, P., Mirrahimi, M., Amini, H., Brune, M., et al.: Real-time quantum feedback prepares and stabilizes photon number states. Nature 477, 73–77 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    Dotsenko, I., Mirrahimi, M., Brune, M., Haroche, S., Raimond, J.M., Rouchon, P.: Quantum feedback by discrete quantum nondemolition measurements: towards on-demand generation of photon-number states. Phys. Rev. A 80, 013805 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    Wang, X., Schirmer, S.G.: Entanglement generation between distant atoms by Lyapunov control. Phys. Rev. A 80, 042305 (2009)ADSCrossRefGoogle Scholar
  27. 27.
    Dong, D., Petersen, I.R.: Sliding mode control of two-level quantum systems. Automatica 48(5), 725–735 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Shi, Z.C., Zhao, X.L., Yi, X.X.: Robust state transfer with high fidelity in spin-1/2 chains by Lyapunov control. Phys. Rev. A 91, 032301 (2015)ADSCrossRefGoogle Scholar
  29. 29.
    Shi, Z.C., Hou, S.C., Wang, L.C., Yi, X.X.: Preparation of edge states by shaking boundaries. Ann. Phys. 373, 286–297 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Ran, D., Shi, Z.-C., Song, J., Xia, Y.: Speeding up adiabatic passage by adding Lyapunov control. Phys. Rev. A 96, 033803 (2017)ADSCrossRefGoogle Scholar
  31. 31.
    Li, C., Song, J., Xia, Y., Ding, W.: Driving many distant atoms into high-fidelity steady state entanglement via Lyapunov control. Opt. Express 26, 951–962 (2018)ADSCrossRefGoogle Scholar
  32. 32.
    Alpaydin, E.: Introduction to Machine Learning, 2nd edn. MIT Press, Cambridge (2010)zbMATHGoogle Scholar
  33. 33.
    Haykin, S.S.: Neural Networks and Learning Machines, 3rd edn. Pearson, New Jersey (2009)Google Scholar
  34. 34.
    Magesan, E., Gambetta, J.M., Córcoles, A.D., Chow, J.M.: Machine learning for discriminating quantum measurement trajectories and improving readout. Phys. Rev. Lett. 114, 200501 (2015)ADSCrossRefGoogle Scholar
  35. 35.
    Mills, K., Spanner, M., Tamblyn, I.: Deep learning and the Schrödinger equation. Phys. Rev. A 96, 042113 (2017)ADSCrossRefGoogle Scholar
  36. 36.
    Melnikov, A.A., Nautrup, H.P., Krenn, M., Dunjko, V., Tiersch, M., Zeilinger, A., Briegel, H.J.: Active learning machine learns to create new quantum experiments. Proc. Natl. Acad. Sci. 115(6), 1221–1226 (2018)ADSCrossRefGoogle Scholar
  37. 37.
    Torlai, G., Mazzola, G., Carrasquilla, J., Troyer, M., Melko, R., Carleo, G.: Neural-network quantum state tomography. Nat. Phys. 14, 447–450 (2018)CrossRefGoogle Scholar
  38. 38.
    Carleo, G., Troyer, M.: Solving the quantum many-body problem with artificial neural network. Science 355, 602–906 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Deng, D.-L.: Machine learning detection of bell nonlocality in quantum many-body systems. Phys. Rev. Lett. 120, 240402 (2018)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Gao, J., Qiao, L.-F., Jiao, Z.-Q., Ma, Y.-C., Hu, C.-Q., Ren, R.-J., Yang, A.-L., Tang, H., Yung, M.-H., Jin, X.-M.: Experimental machine learning of quantum states. Phys. Rev. Lett. 120, 240501 (2018)ADSCrossRefGoogle Scholar
  41. 41.
    Zahedinejad, E., Ghosh, J., Sanders, B.C.: Designing high-fidelity single-shot three-qubit gates: a machine-learning approach. Phys. Rev. Appl. 6, 054005 (2016)ADSCrossRefGoogle Scholar
  42. 42.
    Mavadia, S., Frey, V., Sastrawan, J., Dona, S., Biercuk, M.J.: Prediction and real-time compensation of qubit decoherence via machine learning. Nat. Commun. 8, 14106 (2017)ADSCrossRefGoogle Scholar
  43. 43.
    August, M., Ni, X.: Using recurrent neural networks to optimize dynamical decoupling for quantum memory. Phys. Rev. A 95, 012335 (2017)ADSCrossRefGoogle Scholar
  44. 44.
    Yang, X.-C., Yung, M.-H., Wang, X.: Neural-network-designed pulse sequences for robust control of singlet-triplet qubits. Phys. Rev. A 97, 042324 (2018)ADSCrossRefGoogle Scholar
  45. 45.
    Specht, D.F.: A general regression neural network. IEEE Trans. Neural Netw. 2(6), 568–576 (1991)CrossRefGoogle Scholar
  46. 46.
    Leung, M.T., Chen, A.S., Daouk, H.: Forecasting exchange rates using general regression neural networks. Comput. Oper. Res. 27, 1093–1110 (2000)zbMATHCrossRefGoogle Scholar
  47. 47.
    Li, C., Bovik, A.C., Wu, X.: Blind image quality assessment using a general regression neural network. IEEE Trans. Neural Netw. 22(5), 793–799 (2011)CrossRefGoogle Scholar
  48. 48.
    Liu, J., Bao, W., Shi, L., Zuo, B., Gao, W.: General regression neural network for prediction of sound absorption coefficients of sandwich structure nonwoven absorbers. Appl. Acoust. 76, 128–137 (2014)CrossRefGoogle Scholar
  49. 49.
    Panda, B.N., Bahubalendruni, M.R., Biswal, B.B.: A general regression neural network approach for the evaluation of compressive strength of FDM prototypes. Neural Comput. Appl. 26, 1129–1136 (2015)CrossRefGoogle Scholar
  50. 50.
    Życzkowski, K., Sommers, H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A Math. Gen. 34, 7111 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Center for Quantum Sciences and School of PhysicsNortheast Normal UniversityChangchunChina

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