Solving quantum optimal control problems by wavelets method

Document Type : Research Article


1 Department of Mathematical Sciences, Yazd University, Yazd, Iran.

2 Department of Computer Science, University of Mohaghegh Ardabili, Ardabil, Iran.


We present the quantum equation and synthesize an optimal control proce dure for this equation. We develop a theoretical method for the analysis of quantum optimal control system given by the time depending Schrödinger equation. The Legendre wavelet method is proposed for solving this problem. This can be used as an efficient and accurate computational method for obtaining numerical solutions of different quantum optimal control problems. The distinguishing feature of this paper is that it makes the method, previously used to solve non-quantum control equations based on Legendre wavelets, usable by using a change of variables for quantum control equations.


Main Subjects

1. Albertini, F. and D’Alessandro, D. Time optimal simultaneous control of two level quantum systems, Automatica, 74 (2016), 55–62.
2. Boggess, A. and Narcowich, F.J. A first course in wavelets with Fourier analysis, John Wiley & Sons, 2015.
3. Borzı, A., Salomon, J., and Volkwein, S. Formulation and numerical solution of finite-level quantum optimal control problems, J. Comput. Appl. Math. 216(1) (2008), 170–197.
4. Boscain, U., Charlot, G., Gauthier, J., Guérin, S., and Jauslin, H. Optimal control in laser-induced population transfer for two-and three-level quantum systems, J. Math. Phys. 43(5) (2002), 2107–2132.

5. Chiu, T.Y. and Lin, K.T. Optimal control of two-qubit quantum gates in a non-Markovian open system, 2016 12th IEEE International Conference on Control and Automation (ICCA), (2016), 791–796.
6. Clark, W., Bloch, A., Colombo, L., and Rooney, P. Optimal control of quantum purity for n = 2 systems, 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017), 1317–1322.
7. Cong, Sh. Control of quantum systems: theory and methods, John Wiley & Sons, 2014.
8. Cong, Sh., Wen, J., and Zou, X. Comparison of time optimal control for two level quantum systems, J. Sys. Eng. Electron. 25(1) (2014), 95–103.
9. Dadashi, M.R., Haghighi, A.R., Soltanian, F., and Yari, A. On the numerical solution of optimal control problems via Bell polynomials basis, Iran. J. Numer. Anal. Optim. 10(2) (2020), 197–221.
10. D’alessandro, D. Introduction to quantum control and dynamics. Chap man & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL, 2008.
11. D’alessandro, D. and Dahleh, M. Optimal control of two-level quantum systems, IEEE Transactions on Automatic Control, 46(6) (2001), 866–876.
12. Edrisi-Tabri, Y., Lakestani, M., and Heydari, A. Two numerical methods for nonlinear constrained quadratic optimal control problems using linear B-spline functions, Iran. J. Numer. Anal. Optim. 6(2) (2016), 17–38.
13. Grivopoulos, S. and Bamieh, B. Optimal population transfers for a quantum system in the limit of large transfer time, Proceedings of the 2004 American Control Conference, volume 3, (2004), 2481–2486.
14. Heydari, M.H., Hooshmandasl, M.R., Maalek Ghaini, F.M., and Cattani, C. Wavelets method for solving fractional optimal control problems, Appl. Math. Comput. 286 (2016), 139–154.
15. Itami, T. Nonlinear optimal control as quantum mechanical eigenvalue problems, Automatica, 41(9) (2005), 1617–1622.
16. Jacobs, K. and Shabani, A. Quantum feedback control: how to use verification theorems and viscosity solutions to find optimal protocols, Contemp. Phys. 49(6) (2008), 435–448.
17. Keller, D. Optimal control of a linear stochastic Schrödinger equation, Discrete Contin. Dyn. Syst. 2013, Dynamical systems, differential equations and applications. 9th AIMS Conference. Suppl. 437–446.

18. Koçak, Y., Çelik, E., and Aksoy, N.Y. A note on optimal control problem governed by Schrödinger equation, Open Phys. 13, (2015), 407–413.
19. Kuang, S., Dong, D., and Petersen, I.R. Rapid Lyapunov control of finite dimensional quantum systems, Automatica, 81 (2017), 164–175.
20. Li, J., Yang, X., Peng, X., and Sun, Ch.P. Hybrid quantum-classical approach to quantum optimal control, Phys. Rev. Lett. 118(15) (2017),150503.
21. Lotfi, A., Dehghan, M., and Yousefi, A.A. A numerical technique for solving fractional optimal control problems, Compute. Math. Appl. 62(3)(2011), 1055–1067.
22. Mirrahimi, M., Rouchon, P., and Turinici, G. Lyapunov control of bilinear Schrödinger equations, Automatica, 41(11) (2005), 1987–1994.
23. Mohammadizadeh, F., Tehrani, H.A. and Noori Skandari, M.H. Cheby shev pseudo-spectral method for optimal control problem of Burgers’ equation, Iran. J. Numer. Anal. Optim. 9(2) (2019), 77–102.
24. Oǧuztöreli, M.N., Bellman, R.E., and Oğuztöreli, M.N. Time-lag control systems, Mathematics in Science and Engineering, 24 Academic Press, New York-London 1966.
25. Riviello, G., Brif, C., Long, R., Wu, R.B., Tibbetts, K.M., Ho, T.S., and Rabitz, H. Searching for quantum optimal control fields in the presence of singular critical points, Phys. Rev. A, 90(1) (2014), 013404.
26. Shi, B., Xu, Ch., and Wu, R. Time scaling transformation in quantum optimal control computation, 2018 37th Chinese Control Conference (CCC),2018, pp. 8138–8143.
27. Thirring, W. Classical mathematical physics: dynamical systems and field theories, Springer Science & Business Media, 2013.
28. Toyoglu, F. and Yagubov, G. Numerical solution of an optimal control problem governed by two dimensional Schrödinger equation, Appl. Math. Comput. 4(2) (2015), 30–38.
29. Rehman, M. and Khan, R.A. The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 16(11) (2011), 4163–4173.
30. Damme, L.V., Ansel, Q., Glaser, S.J., and Sugny, D. Robust optimal control of two-level quantum systems, Phys. Rev. A, 95(6) (2017), 13 pp.
31. Wang, Q.F. Quantum optimal control of nonlinear dynamics systems described by Klein-Gordon-Schrödinger equations, Proceeding of American Control Conference, 2006, 1032–1037.

 32. Werschnik, J. and Gross, E. Quantum optimal control theory, J. Phys. B 40(18) (2007), R175–R211.
33. Xu, X. and Xu, D. Legendre wavelets method for approximate solution of fractional-order differential equations under multi-point boundary condi tions, Int. J. Comput. Math. 95(5) (2018), 998–1014.
34. Zhu, W. and Rabitz, H. Attaining optimal controls for manipulating quantum systems, Int. J. Quantum Chem. 93(2) (2003), 50–58.