Krotov method for optimal control of closed quantum systems

O. V. Morzhin ; A. N. Pechen

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 5, pp. 851-908

Résumé

The mathematics of optimal control of quantum systems is of great interest in connection with fundamental problems of physics as well as with existing and prospective applications to quantum technologies. One important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method, which was initially proposed outside of quantum control theory in articles by Krotov and Feldman (1978, 1983). This method was used to develop a novel approach to finding optimal controls for quantum systems in [64] (Tannor, Kazakov, and Orlov, 1992), [65] (Somlói, Kazakov, and Tannor, 1993), and in many other works by various scientists. Our survey discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with different forms of the improvement function (for example, linear or linear-quadratic), different constraints on the control spectrum and on the admissible states of the quantum system, different regularisers, and so on. The survey describes applications of the Krotov method to controlling molecular dynamics and Bose–Einstein condensates, and to quantum gate generation. This method is compared with the GRAPE (GRadient Ascent Pulse Engineering) method, the CRAB (Chopped Random-Basis) method, and the Zhu–Rabitz and Maday–Turinici methods. Bibliography: 158 titles.

Détail

Keywords: quantum control, coherent control, Krotov method, closed quantum systems, quantum technology.

Bibliographie

[1] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, Interscience Publishers John Wiley Sons, Inc., New York–London, 1962, viii+360 pp. | MR | MR | Zbl | Zbl

[2] R. Bellman, Dynamic programming, Princeton Univ. Press, Princeton, NJ, 1957, xxv+342 pp. | MR | MR | Zbl

[3] A. G. Butkovskiy, Yu. I. Samoilenko, Control of quantum-mechanical processes and systems, Math. Appl. (Soviet Ser.), Kluwer Acad. Publ., Dordrecht, 1990, xiv+232 pp. | MR | MR | Zbl

[4] I. V. Krasnov, N. Ya. Shaparev, I. M. Shkedov, Optimalnye lazernye vozdeistviya, Nauka, Novosibirsk, 1989, 93 pp.

[5] S. A. Rice, M. Zhao, Optical control of molecular dynamics, John Wiley Sons, Inc., New York, 2000, xvi+438 pp.

[6] A. D. Bandrauk, M. C. Delfour, C. Le Bris (eds.), Quantum control: mathematical and numerical challenges (Montréal, QC, 2002), CRM Proc. Lecture Notes, 33, Amer. Math. Soc., Providence, RI, 2003, xii+211 pp. | MR | Zbl

[7] D. D'Alessandro, “Directions in the theory of quantum control”, Multidisciplinary research in control (Santa Barbara, CA, 2002), Lect. Notes Control Inf. Sci., 289, Springer, Berlin, 2003, 73–80 | DOI | MR | Zbl

[8] M. Shapiro, P. Brumer, Principles of the quantum control of molecular processes, John Wiley Sons, Inc., Hoboken, NJ, 2003, xiv+354 pp. ; 2nd ed., 2012, 544 pp. | DOI

[9] D. J. Tannor, Introduction to quantum mechanics: a time-dependent perspective, Univ. Science Books, Sausilito, CA, 2007, 662 pp.

[10] V. S. Letokhov, Laser control of atoms and molecules, Oxford Univ. Press, Oxford, 2007, 328 pp.

[11] D. D'Alessandro, Introduction to quantum control and dynamics, Chapman Hall/CRC Appl. Math. Nonlinear Sci. Ser., Chapman Hall/CRC, Boca Raton, FL, 2008, xiv+343 pp. | DOI | MR | Zbl

[12] A. L. Fradkov, Cybernetical physics. From control of chaos to quantum control, Underst. Complex Syst., Springer, Berlin, 2007, xii+241 pp. | DOI | MR | Zbl

[13] C. Brif, R. Chakrabarti, H. Rabitz, “Control of quantum phenomena: past, present and future”, New J. Phys., 12:7 (2010), 075008, 68 pp. | DOI

[14] D. Y. Dong, I. R. Petersen, “Quantum control theory and applications: a survey”, IET Control Theory Appl., 4:12 (2010), 2651–2671 | DOI | MR

[15] H. M. Wiseman, G. J. Milburn, Quantum measurement and control, Cambridge Univ. Press, Cambridge, 2010, xvi+460 pp. | DOI | MR | Zbl

[16] C. Altafini, F. Ticozzi, “Modeling and control of quantum systems: an introduction”, IEEE Trans. Automat. Control, 57:8 (2012), 1898–1917 | DOI | MR | Zbl

[17] B. Bonnard, D. Sugny, Optimal control with applications in space and quantum dynamics, AIMS Ser. Appl. Math., 5, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2012, xvi+283 pp. | MR | Zbl

[18] J. E. Gough, “Introduction: Principles and applications of quantum control engineering”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370:1979 (2012), 5241–5258 | DOI | MR | Zbl

[19] S. Cong, Control of quantum systems. Theory and methods, John Wiley Sons, Hoboken, NJ, 2014, xvii+425 pp. | DOI | Zbl

[20] W. Dong, R. Wu, X. Yuan, C. Li, T.-J. Tarn, “The modelling of quantum control systems”, Sci. Bull., 60:17 (2015), 1493–1508 | DOI

[21] S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, F. K. Wilhelm, “Training Schrödinger's cat: quantum optimal control. Strategic report on current status, visions and goals for research in Europe”, Eur. Phys. J. D, 69:12 (2015), 279, 24 pp. | DOI

[22] C. P. Koch, “Controlling open quantum systems: tools, achievements, and limitations”, J. Phys. Condens. Matter, 28:21 (2016), 213001, 13 pp. | DOI

[23] A. Borzì, G. Ciaramella, M. Sprengel, Formulation and numerical solution of quantum control problems, Comput. Sci. Eng., 16, SIAM, Philadelphia, PA, 2017, x+390 pp. | DOI | MR | Zbl

[24] V. P. Belavkin, “Theory of the control of observable quantum systems”, Autom. Remote Control, 44:2 (1983), 178–188 | MR | Zbl

[25] H. M. Wiseman, G. J. Milburn, “Quantum theory of optical feedback via homodyne detection”, Phys. Rev. Lett., 70:5 (1993), 548–551 | DOI

[26] R. S. Judson, H. Rabitz, “Teaching lasers to control molecules”, Phys. Rev. Lett., 68:10 (1992), 1500–1503 | DOI

[27] A. Pechen, H. Rabitz, “Teaching the environment to control quantum systems”, Phys. Rev. A, 73:6 (2006), 062102, 6 pp. | DOI

[28] O. V. Morzhin, A. N. Pechen, “Maximization of the overlap between density matrices for a two-level open quantum system driven by coherent and incoherent controls”, Lobachevskii J. Math., 40:10 (2019) (in print)

[29] U. Boscain, G. Charlot, J.-P. Gauthier, S. Guérin, H.-R. Jauslin, “Optimal control in laser-induced population transfer for two- and three-level quantum systems”, J. Math. Phys., 43:5 (2002), 2107–2132 | DOI | MR | Zbl

[30] U. Boscain, T. Chambrion, G. Charlot, “Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimal energy”, Discrete Contin. Dyn. Syst. Ser. B, 5:4 (2005), 957–990 | DOI | MR | Zbl

[31] U. Boscain, P. Mason, “Time minimal trajectories for a spin 1/2 particle in a magnetic field”, J. Math. Phys., 47:6 (2006), 062101, 29 pp. | DOI | MR | Zbl

[32] A. Carlini, A. Hosoya, T. Koike, Y. Okudaira, “Time optimal quantum evolution”, Phys. Rev. Lett., 96:6 (2006), 060503 | DOI

[33] P. Salamon, K. H. Hoffmann, A. Tsirlin, “Optimal control in a quantum cooling problem”, Appl. Math. Lett., 25:10 (2012), 1263–1266 | DOI | MR | Zbl

[34] E. Assémat, M. Lapert, D. Sugny, S. J. Glaser, “On the application of geometric optimal control theory to nuclear magnetic resonance”, Math. Control Relat. Fields, 3:4 (2013), 375–396 | DOI | MR | Zbl

[35] U. Boscain, F. Grönberg, R. Long, H. Rabitz, “Minimal time trajectories for two-level quantum systems with two bounded controls”, J. Math. Phys., 55:6 (2014), 062106, 25 pp. | DOI | MR | Zbl

[36] R. Romano, “Geometric analysis of minimum time trajectories for a two level quantum system”, Phys. Rev. A, 90:6 (2014), 062302 | DOI

[37] F. Albertini, D. D'Alessandro, “Time optimal simultaneous control of two level quantum systems”, Automatica, 74 (2016), 55–62 | DOI | MR | Zbl

[38] T. Schulte-Herbrüggen, S. J. Glaser, G. Dirr, U. Helmke, “Gradient flows for optimization in quantum information and quantum dynamics: foundations and applications”, Rev. Math. Phys., 22:6 (2010), 597–667 | DOI | MR | Zbl

[39] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, S. J. Glaser, “Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms”, J. Magn. Reson., 172:2 (2005), 296–305 | DOI

[40] G. Jäger, D. M. Reich, M. H. Goerz, C. P. Koch, U. Hohenester, “Optimal quantum control of Bose–Einstein condensates in magnetic microtraps: Comparison of gradient-ascent-pulse-engineering and Krotov optimization schemes”, Phys. Rev. A, 90:3 (2014), 033628, 9 pp. | DOI

[41] T. Caneva, T. Calarco, S. Montangero, “Chopped random-basis quantum optimization”, Phys. Rev. A, 84:2 (2011), 022326 | DOI

[42] W. Zhu, H. Rabitz, “A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator”, J. Chem. Phys., 109:2 (1998), 385–391 | DOI

[43] Y. Maday, G. Turinici, “New formulations of monotonically convergent quantum control algorithms”, J. Chem. Phys., 118:18 (2003), 8191–8196 | DOI

[44] J. Gough, V. P. Belavkin, O. G. Smolyanov, “Hamilton–Jacobi–Bellman equations for quantum optimal feedback control”, J. Opt. B Quantum Semiclass. Opt., 7:10 (2005), S237–S244 | DOI | MR

[45] A. Pechen, A. Trushechkin, “Measurement-assisted Landau–Zener transitions”, Phys. Rev. A, 91:5 (2015), 052316, 15 pp. | DOI

[46] M. K. Riahi, J. Salomon, S. J. Glaser, D. Sugny, “Fully efficient time-parallelized quantum optimal control algorithm”, Phys. Rev. A, 93:4 (2016), 043410 | DOI

[47] M. S. Anan'evskii, A. L. Fradkov, “Control of the observables in the finite-level quantum systems”, Autom. Remote Control, 66:5 (2005), 734–745 | DOI | MR | Zbl

[48] A. N. Pechen, “On the speed gradient method for generating unitary quantum operations for closed quantum systems”, Russian Math. Surveys, 71:3 (2016), 597–599 | DOI | DOI | MR | Zbl

[49] T.-S. Ho, H. Rabitz, “Accelerated monotonic convergence of optimal control over quantum dynamics”, Phys. Rev. E (3), 82:2 (2010), 026703, 15 pp. | DOI | MR

[50] O. V. Morzhin, A. N. Pechen, “Minimal time generation of density matrices for a two-level quantum system driven by coherent and incoherent controls”, Int. J. Theor. Phys., first online 2019, 9 pp., DOI: 10.1007/s10773-019-04149-w | DOI

[51] D. Dong, C. Chen, T.-J. Tarn, A. Pechen, H. Rabitz, “Incoherent control of quantum systems with wavefunction-controllable subspaces via quantum reinforcement learning”, IEEE Trans. Systems Man Cybernet. Part B (Cybernet.), 38:4 (2008), 957–962 | DOI

[52] M. Y. Niu, S. Boixo, V. Smelyanskiy, H. Neven, Universal quantum control through deep reinforcement learning, 2018, 21 pp., arXiv: 1803.01857

[53] J. P. P. Zauleck, R. de Vivie-Riedle, “Constructing grids for molecular quantum dynamics using an autoencoder”, J. Chem. Theory Comput., 14:1 (2018), 55–62 | DOI

[54] P. Palittapongarnpim, B. C. Sanders, “Enter the machine”, Nat. Phys., 14:5 (2018), 432–433 | DOI

[55] G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, G. Carleo, “Neural-network quantum state tomography”, Nat. Phys., 14:5 (2018), 447–450 | DOI

[56] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, S. Lloyd, “Quantum machine learning”, Nature, 549:7671 (2017), 195–202 | DOI

[57] V. F. Krotov, I. N. Feldman, “Iteratsionnye metody resheniya ekstremalnykh zadach”, Modelirovanie tekhniko-ekonomicheskikh protsessov, Izd-vo MESI, M., 1978, 22–35

[58] V. F. Krotov, I. N. Fel'dman, “An iterative method for solving optimal control problems”, Engrg. Cybernetics, 21:2 (1983), 123–130 | MR

[59] V. F. Krotov, V. I. Gurman, Metody i zadachi optimalnogo upravleniya, Nauka, M., 1973, 446 pp. | MR | Zbl

[60] V. F. Krotov, Global methods in optimal control theory, Monogr. Textbooks Pure Appl. Math., 195, Marcel Dekker, Inc., New York, 1996, xvi+384 pp. | MR | Zbl

[61] A. I. Konnov, V. F. Krotov, “On global methods of successive improvement of control processes”, Autom. Remote Control, 60:10 (1999), 1427–1436 | MR | Zbl

[62] V. A. Kazakov, V. F. Krotov, “Optimal control of resonant interaction between light and matter”, Autom. Remote Control, 48:4 (1987), 430–434 | MR

[63] V. F. Krotov, “Global methods to improve control and optimal control of resonance interaction of light and matter”, Modeling and control of systems in engineering, quantum mechanics, economics and biosciences (Sophia-Antipolis, 1988), Lect. Notes Control Inf. Sci., 121, Springer, Berlin, 1989, 267–298 | DOI | MR | Zbl

[64] D. J. Tannor, V. Kazakov, V. Orlov, “Control of photochemical branching: novel procedures for finding optimal pulses and global upper bounds”, Time-dependent quantum molecular dynamics, Nato ASI Ser. Ser. B, 299, Springer, Boston, MA, 1992, 347–360 | DOI

[65] J. Somlói, V. A. Kazakov, D. J. Tannor, “Controlled dissociation of $I_2$ via optical transitions between the X and B electronic states”, Chem. Phys., 172:1 (1993), 85–98 | DOI

[66] S. E. Sklarz, D. J. Tannor, “Loading a Bose–Einstein condensate onto an optical lattice: an application of optimal control theory to the nonlinear Schrödinger equation”, Phys. Rev. A, 66:5 (2002), 053619, 9 pp. | DOI

[67] J. P. Palao, R. Kosloff, C. P. Koch, “Protecting coherence in optimal control theory: state-dependent constraint approach”, Phys. Rev. A, 77:6 (2008), 063412, 11 pp. | DOI

[68] T. Szakács, B. Amstrup, P. Gross, R. Kosloff, H. Rabitz, A. Lörincz, “Locking a molecular bond: a case study of CsI”, Phys. Rev. A, 50:3 (1994), 2540–2547 | DOI

[69] I. R. Solá, J. Santamaria, D. J. Tannor, “Optimal control of multiphoton excitation: a black box or a flexible toolkit?”, J. Phys. Chem. A, 102:23 (1998), 4301–4309 | DOI

[70] A. Bartana, R. Kosloff, D. J. Tannor, “Laser cooling of molecules by dynamically trapped states”, Chem. Phys., 267:1-3 (2001), 195–207 | DOI

[71] C. P. Koch, J. P. Palao, R. Kosloff, F. Masnou-Seeuws, “Stabilization of ultracold molecules using optimal control theory”, Phys. Rev. A, 70:1 (2004), 013402, 14 pp. | DOI

[72] T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, G. E. Santoro, “Optimal control at the quantum speed limit”, Phys. Rev. Lett., 103:24 (2009), 240501 | DOI

[73] M. Ndong, C. P. Koch, “Vibrational stabilization of ultracold KRb molecules: a comparative study”, Phys. Rev. A, 82:4 (2010), 043437, 10 pp. | DOI

[74] R. Eitan, M. Mundt, D. J. Tannor, “Optimal control with accelerated convergence: combining the Krotov and quasi-Newton methods”, Phys. Rev. A, 83:5 (2011), 053426, 10 pp. | DOI

[75] P. Kumar, S. A. Malinovskaya, V. S. Malinovsky, “Optimal control of population and coherence in three-level $\Lambda$ systems”, J. Phys. B, 44:15 (2011), 154010, 10 pp. | DOI

[76] J. P. Palao, D. M. Reich, C. P. Koch, “Steering the optimization pathway in the control landscape using constraints”, Phys. Rev. A, 88:5 (2013), 053409, 8 pp. | DOI

[77] E. A. Trushkova, “Metod globalnogo uluchsheniya dlya gamiltonovykh sistem s upravlyaemymi koeffitsientami”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:1(2) (2013), 95–99 | Zbl

[78] M. Ndong, C. P. Koch, D. Sugny, “Time optimization and state-dependent constraints in the quantum optimal control of molecular orientation”, J. Modern Opt., 61:10 (2014), 857–863 | DOI | MR | Zbl

[79] M. Goerz, Optimizing robust quantum gates in open quantum systems, Diss. ... Dr. rer. Nat., Univ. Kassel, 2015, xvi+221\par pp. https://michaelgoerz.net/research/diss_goerz_letter.pdf

[80] J. P. Palao, R. Kosloff, “Quantum computing by an optimal control algorithm for unitary transformations”, Phys. Rev. Lett., 89:18 (2002), 188301, 13 pp. | DOI

[81] J. P. Palao, R. Kosloff, “Optimal control theory for unitary transformations”, Phys. Rev. A, 68:6 (2003), 062308 | DOI

[82] P. Treutlein, T. W. Hänsch, J. Reichel, A. Negretti, M. A. Cirone, T. Calarco, “Microwave potentials and optimal control for robust quantum gates on an atom chip”, Phys. Rev. A, 74:2 (2006), 022312, 13 pp. | DOI

[83] G. de Chiara, T. Calarco, M. Anderlini, S. Montangero, P. J. Lee, B. L. Brown, W. D. Phillips, J. V. Porto, “Optimal control of atom transport for quantum gates in optical lattices”, Phys. Rev. A, 77:5 (2008), 052333, 9 pp. | DOI

[84] C. Gollub, M. Kowalewski, R. de Vivie-Riedle, “Monotonic convergent optimal control theory with strict limitations on the spectrum of optimized laser fields”, Phys. Rev. Lett., 101:7 (2008), 073002, 4 pp. | DOI

[85] T. Koike, Y. Okudaira, “Time complexity and gate complexity”, Phys. Rev. A, 82:4 (2010), 042305 | DOI

[86] K. Singer, U. Poschinger, M. Murphy, P. Ivanov, F. Ziesel, T. Calarco, F. Schmidt-Kaler, “Colloquium: Trapped ions as quantum bits: essential numerical tools”, Rev. Mod. Phys., 82:3 (2010), 2609–2632 | DOI

[87] M. H. Goerz, T. Calarco, C. P. Koch, “The quantum speed limit of optimal controlled phasegates for trapped neutral atoms”, J. Phys. B, 44:15 (2011), 154011, 10 pp. | DOI

[88] M. M. Müller, D. M. Reich, M. Murphy, H. Yuan, J. Vala, K. B. Whaley, T. Calarco, C. P. Koch, “Optimizing entangling quantum gates for physical systems”, Phys. Rev. A, 84:4 (2011), 042315, 8 pp. | DOI

[89] D. M. Reich, M. Ndong, C. P. Koch, “Monotonically convergent optimization in quantum control using Krotov's method”, J. Chem. Phys., 136:10 (2012), 104103, 16 pp. | DOI

[90] M. H. Goerz, G. Gualdi, D. M. Reich, C. P. Koch, F. Motzoi, K. B. Whaley, J. Vala, M. M. Müller, S. Montangero, T. Calarco, “Optimizing for an arbitrary perfect entangler. II. Application”, Phys. Rev. A, 91:6 (2015), 062307, 11 pp. | DOI | MR

[91] M. H. Goerz, K. B. Whaley, C. P. Koch, “Hybrid optimization schemes for quantum control”, EPJ Quantum Technol., 2 (2015), 21 | DOI

[92] D. Basilewitsch, R. Schmidt, D. Sugny, S. Maniscalco, C. P. Koch, “Beating the limits with initial correlations”, New J. Phys., 19 (2017), 113042, 16 pp. | DOI

[93] M. H. Goerz, F. Motzoi, K. B. Whaley, C. P. Koch, “Charting the circuit QED design landscape using optimal control theory”, npj Quantum Info., 3 (2017), 37, 10 pp. | DOI

[94] D. Basilewitsch, L. Marder, C. P. Koch, “Dissipative quantum dynamics and optimal control using iterative time ordering: an application to superconducting qubits”, Eur. Phys. J. B, 91:7 (2018), 161, 16 pp. | DOI | MR

[95] M. H. Goerz, K. Jacobs, “Efficient optimization of state preparation in quantum networks using quantum trajectories”, Quantum Sci. Technol., 3:4 (2018), 045005 | DOI

[96] D. Basilewitsch, F. Cosco, N. Lo Gullo, M. Möttönen, T. Ala-Nissilä, C. P. Koch, S. Maniscalco, Reservoir engineering using quantum optimal control for qubit reset, 2019, 9 pp., arXiv: 1903.05059

[97] G. Jäger, Optimal quantum control of Bose–Einstein condensates, Thesis (Dr. rer. Nat.), Univ. Graz, 2015, 107 pp.

[98] J. J. W. H. Sørensen, M. O. Aranburu, T. Heinzel, J. F. Sherson, “Quantum optimal control in a chopped basis: applications in control of Bose–Einstein condensates”, Phys. Rev. A, 98:2 (2018), 022119 | DOI

[99] I. I. Maximov, Z. Tošner, N. C. Nielsen, “Optimal control design of NMR and dynamic nuclear polarization experiments using monotonically convergent algorithms”, J. Chem. Phys., 128:18 (2008), 184505, 14 pp. | DOI

[100] I. I. Maximov, J. Salomon, G. Turinici, N. C. Nielsen, “A smoothing monotonic convergent optimal control algorithm for nuclear magnetic resonance pulse sequence design”, J. Chem. Phys., 132:8 (2010), 084107 | DOI

[101] M. S. Vinding, I. I. Maximov, Z. Tošner, N. C. Nielsen, “Fast numerical design of spatial-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods”, J. Chem. Phys., 137:5 (2012), 054203 | DOI

[102] Programmnye sredstva, soderzhaschie realizatsii metoda Krotova , https://www.qdyn-library.net/https://github.com/qucontrol/krotov

[103] M. H. Goerz, D. Basilewitsch, F. Gago-Encinas, M. G. Krauss, K. P. Horn, D. M. Reich, C. P. Koch, Krotov: A Python implementation of Krotov's method for quantum optimal control, 2019, 26 pp., arXiv: 1902.11284

[104] D. M. Reich, Efficient characterisation and optimal control of open quantum systems – mathematical foundations and physical applications, Diss. ... Dr. rer. Nat., Univ. Kassel, 2015, 178 pp. https://kobra.uni-kassel.de/handle/123456789/2015061948576

[105] R. Kosloff, S. A. Rice, P. Gaspard, S. Tersigni, D. J. Tannor, “Wavepacket dancing: achieving chemical selectivity by shaping light pulses”, Chem. Phys., 139:1 (1989), 201–220 | DOI

[106] M. Lapert, R. Tehini, G. Turinici, D. Sugny, “Monotonically convergent optimal control theory of quantum systems under a nonlinear interaction with the control field”, Phys. Rev. A, 78:2 (2008), 023408 | DOI

[107] V. F. Krotov, “Quantum system control optimization”, Dokl. Math., 78:3 (2008), 949–952 | DOI | MR | Zbl

[108] V. F. Krotov, “Control of the quantum systems and some ideas of the optimal control theory”, Autom. Remote Control, 70:3 (2009), 357–365 | DOI | MR | Zbl

[109] K. Sundermann, R. de Vivie-Riedle, “Extensions to quantum optimal control algorithms and applications to special problems in state selective molecular dynamics”, J. Chem. Phys., 110:4 (1999), 1896–1904 | DOI

[110] S. G. Schirmer, P. de Fouquieres, “Efficient algorithms for optimal control of quantum dynamics: the Krotov method unencumbered”, New J. Phys., 13 (2011), 073029, 35 pp. | DOI

[111] N. Boussaïd, M. Caponigro, T. Chambrion, “Periodic control laws for bilinear quantum systems with discrete spectrum”, Proc. Amer. Control Conf. (ACC), 2012, 5819–5824 | DOI

[112] A. Yu. Kitaev, “Quantum computations: algorithms and error correction”, Russian Math. Surveys, 52:6 (1997), 1191–1249 | DOI | DOI | MR | Zbl

[113] A. Yu. Kitaev, A. H. Shen, M. N. Vyalyi, Classical and quantum computation, Grad. Stud. Math., 47, Amer. Math. Soc., Providence, RI, 2002, xiv+257 pp. | DOI | MR | Zbl

[114] K. A. Valiev, “Quantum computers and quantum computations”, Phys. Usp., 48:1 (2005), 1–36 | DOI | DOI

[115] M. A. Nielsen, I. L. Chuang, Quantum computation and quantum information, 10th ann. ed., Cambridge Univ. Press, Cambridge, 2010, xxxi+676 pp. | DOI | MR | Zbl

[116] M. Ohya, I. Volovich, Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems, Theoret. Math. Phys., Springer, Dordrecht, 2011, xx+759 pp. | DOI | MR | Zbl

[117] A. S. Holevo, Quantum systems, channels, information. A mathematical introduction, De Gruyter Stud. Math. Phys., 16, De Gruyter, Berlin, 2012, xiv+349 pp. | DOI | MR | Zbl

[118] C. Jurdjevic, H. Sussmann, “Control systems on Lie groups”, J. Differential Equations, 12:2 (1972), 313–329 | DOI | MR | Zbl

[119] A. Pechen, N. Il'in, “Trap-free manipulation in the Landau–Zener system”, Phys. Rev. A, 86:5 (2012), 052117, 6 pp. | DOI

[120] A. N. Pechen, N. B. Il'in, “Coherent control of a qubit is trap-free”, Proc. Steklov Inst. Math., 285 (2014), 233–240 | DOI | DOI | MR | Zbl

[121] S. van Frank, M. Bonneau, J. Schmiedmayer, S. Hild, C. Gross, M. Cheneau, I. Bloch, T. Pichler, A. Negretti, T. Calarco, S. Montangero, “Optimal control of complex atomic quantum systems”, Sci. Rep., 6 (2016), 34187 | DOI

[122] U. Hohenester, P. K. Rekdal, A. Borzì, J. Schmiedmayer, “Optimal quantum control of Bose–Einstein condensates in magnetic microtraps”, Phys. Rev. A, 75:2 (2007), 023602 | DOI

[123] R. Bücker, T. Berrada, S. van Frank, J.-F. Schaff, T. Schumm, J. Schmiedmayer, G. Jäger, J. Grond, U. Hohenester, “Vibrational state inversion of a Bose–Einstein condensate: optimal control and state tomography”, J. Phys. B, 46:10 (2013), 104012 | DOI

[124] M. Hintermüller, D. Marahrens, P. A. Markowich, C. Sparber, “Optimal bilinear control of Gross–Pitaevskii equations”, SIAM J. Control Optim., 51:3 (2013), 2509–2543 | DOI | MR | Zbl

[125] S. van Frank, A. Negretti, T. Berrada, R. Bücker, S. Montangero, J.-F. Schaff, T. Schumm, T. Calarco, J. Schmiedmayer, “Interferometry with non-classical motional states of a Bose–Einstein condensate”, Nature Comm., 5 (2014), 4009, 6 pp. | DOI

[126] D. Hocker, J. Yan, H. Rabitz, “Optimal nonlinear coherent mode transitions in Bose–Einstein condensates utilizing spatiotemporal controls”, Phys. Rev. A, 93:5 (2016), 053612 | DOI

[127] A. G. Butkovskii, Yu. I. Samoilenko, “Controllability of quantum objects”, Soviet Phys. Dokl., 25 (1980), 22–24 | MR | Zbl

[128] G. M. Huang, T. J. Tarn, J. W. Clark, “On the controllability of quantum-mechanical systems”, J. Math. Phys., 24:11 (1983), 2608–2618 | DOI | MR | Zbl

[129] C. Altafini, “Controllability of quantum mechanical systems by root space decomposition of $\mathfrak{su}(n)$”, J. Math. Phys., 43:5 (2002), 2051–2062 | DOI | MR | Zbl

[130] G. Turinici, H. Rabitz, “Wavefunction controllability for finite-dimensional bilinear quantum systems”, J. Phys. A, 36:10 (2003), 2565–2576 | DOI | MR | Zbl

[131] I. Kurniawan, G. Dirr, U. Helmke, “Controllability aspects of quantum dynamics: a unified approach for closed and open systems”, IEEE Trans. Automat. Control, 57:8 (2012), 1984–1996 | DOI | MR | Zbl

[132] D. D'Alessandro, F. Albertini, R. Romano, “Exact algebraic conditions for indirect controllability of quantum systems”, SIAM J. Control Optim., 53:3 (2015), 1509–1542 | DOI | MR | Zbl

[133] U. Boscain, J.-P. Gauthier, F. Rossi, M. Sigalotti, “Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems”, Comm. Math. Phys., 333:3 (2015), 1225–1239 | DOI | MR | Zbl

[134] A. Agrachev, U. Boscain, J.-P. Gauthier, M. Sigalotti, “A note on time-zero controllability and density of orbits for quantum systems”, Proceedings of the IEEE 56th annual conference on decision and control (Melbourne, VIC, 2017), IEEE Conf. Decis. Control, 56, 2017, 5535–5538 | DOI

[135] H. Fu, S. G. Schirmer, A. I. Solomon, “Complete controllability of finite-level quantum systems”, J. Phys. A, 34:8 (2001), 1679–1690 | DOI | MR | Zbl

[136] F. Albertini, D. D'Alessandro, “Notions of controllability for bilinear multilevel quantum systems”, IEEE Trans. Automat. Control, 48:8 (2003), 1399–1403 | DOI | MR | Zbl

[137] H. A. Rabitz, M. M. Hsieh, C. M. Rosenthal, “Quantum optimally controlled transition landscapes”, Science, 303:5666 (2004), 1998–2001 | DOI

[138] K. W. Moore, H. Rabitz, “Exploring constrained quantum control landscapes”, J. Chem. Phys., 137:13 (2012), 134113, 16 pp. | DOI

[139] G. Riviello, C. Brif, R. Long, R.-B. Wu, K. Moore Tibbetts, T.-S. Ho, H. Rabitz, “Searching for quantum optimal control fields in the presence of singular critical points”, Phys. Rev. A, 90:1 (2014), 013404 | DOI

[140] A. N. Pechen, D. J. Tannor, “Control of quantum transmission is trap free”, Canadian J. Chem., 92:2 (2014), 157–159 | DOI

[141] A. N. Pechen, D. J. Tannor, “Are there traps in quantum control landscapes?”, Phys. Rev. Lett., 106:12 (2011), 120402 | DOI

[142] P. de Fouquieres, S. G. Schirmer, “A closer look at quantum control landscapes and their implication for control optimization”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 16:3 (2013), 1350021, 24 pp. | DOI | MR | Zbl

[143] H. Rabitz, T.-S. Ho, R. Long, R. Wu, C. Brif, “Comment on ‘Are there traps in quantum control landscapes?’ ”, Phys. Rev. Lett., 108:19 (2012), 198901 | DOI

[144] A. N. Pechen, N. B. Il'in, “On critical points of the objective functional for maximization of qubit observables”, Russian Math. Surveys, 70:4 (2015), 782–784 | DOI | DOI | MR | Zbl

[145] N. B. Il'in, A. N. Pechen, “Conditions for the absence of local extrema in problems of quantum coherent control”, Proc. Steklov Inst. Math., 301:1 (2018), 109–113 | DOI | DOI | MR | Zbl

[146] V. A. Srochko, Iteratsionnye metody resheniya zadach optimalnogo upravleniya, Fizmatlit, M., 2000, 160 pp.

[147] V. F. Krotov, A. V. Bulatov, O. V. Baturina, “Optimization of linear systems with controllable coefficients”, Autom. Remote Control, 72:6 (2011), 1199–1212 | DOI | MR | Zbl

[148] A. F. Filippov, Differential equations with discontinuous righthand sides, Math. Appl. (Soviet Ser.), 18, Kluwer Acad. Publ., Dordrecht, 1988, x+304 pp. | DOI | MR | MR | Zbl | Zbl

[149] V. I. Gurman, V. A. Baturin, E. V. Danilina i dr. (red.), Novye metody uluchsheniya upravlyaemykh protsessov, (IrVTs SO AN SSSR), Nauka, Novosibirsk, 1987, 184 pp. | MR | Zbl

[150] O. V. Baturina, O. V. Morzhin, “Optimal control of the spin system on a basis of the global improvement method”, Autom. Remote Control, 72:6 (2011), 1213–1220 | DOI | MR | Zbl

[151] V. F. Krotov, O. V. Morzhin, E. A. Trushkova, “Discontinuous solutions of the optimal control problems. Iterative optimization method”, Autom. Remote Control, 74:12 (2013), 1948–1968 | DOI | MR | Zbl

[152] Y. Ohtsuki, G. Turinici, H. Rabitz, “Generalized monotonically convergent algorithms for solving quantum optimal control problems”, J. Chem. Phys., 120:12 (2004), 5509–5517 | DOI

[153] P. Gross, D. Neuhauser, H. Rabitz, “Optimal control of curve-crossing systems”, J. Chem. Phys., 96:4 (1992), 2834–2845 | DOI

[154] H. Jirari, F. W. J. Hekking, O. Buisson, “Optimal control of superconducting $N$-level quantum systems”, Europhys. Lett. EPL, 87:2 (2009), 28004, 6 pp. | DOI

[155] E. A. Trushkova, “Global control improvement algorithms”, Autom. Remote Control, 72:6 (2011), 1282–1290 | DOI | MR | Zbl

[156] K. E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge Monogr. Appl. Comput. Math., 4, Cambridge Univ. Press, Cambridge, 1997, xvi+552 pp. | DOI | MR | Zbl

[157] M. Ndong, H. Tal-Ezer, R. Kosloff, C. P. Koch, “A Chebychev propagator for inhomogeneous Schrödinger equations”, J. Chem. Phys., 130:12 (2009), 124108, 12 pp. | DOI

[158] P. de Fouquieres, S. G. Schirmer, S. J. Glaser, I. Kuprov, “Second order gradient ascent pulse engineering”, J. Magn. Reson., 212:2 (2011), 412–417 | DOI