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Quantum optimal control using the adjoint method


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Control of quantum systems is central in a variety of present and perspective applications ranging from quantum optics and quantum chemistry to semiconductor nanostructures, including the emerging fields of quantum computation and quantum communication. In this paper, a review of recent developments in the field of optimal control of quantum systems is given with a focus on adjoint methods and their numerical implementation. In addition, the issues of exact controllability and optimal control are discussed for finite- and infinitedimensional quantum systems. Some insight is provided considering ’two-level’ models. This work is completed with an outlook to future developments.


  • Institut für Mathematik, Universität Würzburg
    Emil-Fischer-Strasse 30, 97074 Würzburg, Germany


  • C. Altafini, A.M. Bloch, M.R. James, A. Loria, and P. Rouchon, Special Issue on Control of Quantum Mechanical Systems, IEEE Transactions on Automatic Control 57, 1893 (2012). [Crossref]
  • M. Annunziato and A. Borzì, Optimal control of probability density functions of stochastic processes, Mathematical Modelling and Analysis 15, 393 (2010).
  • M. Annunziato and A. Borzì, A Fokker-Planck control framework for multidimensional stochastic processes, Journal of Computational and Applied Mathematics 237, 487 (2013).
  • M. Annunziato and A. Borzì, A Fokker-Planck-based control of a two-level open quantum system, submitted.
  • X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys. 4, 729 (2008).
  • A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses, Science 282, 919 (1998).
  • A. Auger, A. Ben Haj Yedder, E. Cances, C. Le Bris, C.M. Dion, A. Keller, and O. Atabek, Optimal laser control of molecular systems: methodology and results, Math. Models Methods Appl. Sci. 12, 1281 (2002).
  • J.M. Ball, J.E. Marsden, and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Contr. Optim 20, 575 (1982).
  • W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput. 25, 1674 (2004).
  • A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time, Springer, Berlin, (2009).
  • K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, Journal of Functional Analysis 232, 328 (2006).
  • A.M. Bloch, R.W. Brockett, C. Rangan, The Controllability of Infinite Quantum Systems and Closed Subspace Criteria, IEEE Transaction Automatic Control 55, 1897 (2010).
  • A. Borzì and U. Hohennester, Multigrid optimization schemes for solving Bose-Einstein condensate control problems, SIAM J. Scientific Computing 30, 441 (2008).
  • A. Borzì, J. Salomon, and S. Volkwein, Formulation and numerical solution of finite-level quantum optimal control problems, J. Comput. Appl. Math. 216, 170 (2008). [Crossref]
  • A. Borzì, G. Stadler, and U. Hohenester, Optimal quantum control in nanostructures: Theory and application to a generic three-level system, Phys. Rev. A 66, 053811 (2002). [Crossref]
  • U. Boscain, G. Charlot, J.-P. Gauthier, S. Guerin, H.-R. Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems, Journal of Mathematical Physics 43, 2107 (2002). [Crossref]
  • J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI (1999).
  • D. Bouwmeester, A. Ekert, and A. Zeilinger, editors. phThe Physics of Quantum Information. Springer, Berlin, (2000).
  • R. Bücker, J. Grond, S. Manz, T. Berrada, T. Betz, C. Koller, U. Hohenester, T. Schumm, A. Perrin, J. Schmiedmayer, Twin-atom beams, Nature Physics 7, 608 (2011). [Crossref]
  • H.P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, (2007).
  • C. Brif, R. Chakrabarti and H. Rabitz, Control of quantum phenomena: past, present and future, New Journal of Physics 12, 075008 (2010). [Crossref]
  • P.W. Brumer and M. Shapiro, Principles of the Quantum Control of Molecular Processes, Wiley-VCH, Berlin, (2003).
  • J. Burkardt, M. Gunzburger, and J. Peterson, Insensitive functionals, inconsistent gradients, spurious minima, and regularized functionals in flow optimization problems, Int. J. Comput. Fluid Dyn. 16, 171 (2002). [Crossref]
  • A.G. Butkovskiy and Yu.I. Samoilenko, Control of Quantum-Mechanical Processes and Systems, Kluwer Acad. Publ. (1990).
  • A.Z. Capri, Nonrelativistic Quantum Mechanics, World Scientific Publishing Co. Inc., River Edge, NJ, (2002).
  • A. Castro and I. V. Tokatly, Quantum optimal control theory in the linear response formalism, Phys. Rev. A 84, 033410 (2011). [Crossref]
  • T. Chambrion, P. Mason, M. Sigalotti and U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Annales de l’Institut Henri Poincaré - Non Linear Analysis.
  • J.-M. Coron, A. Grigoriu, C. Lefter, and G. Turinici, Quantum control design by lyapunov trajectory tracking for dipole and polarizability coupling, New Journal of Physics 11, 105034 (2009). [Crossref]
  • P. Crouch and F. Silva Leite, Controllability on classical Lie groups, Mathematics of Control Signals and Systems 1, 31 (1988).
  • Y.H. Dai and Y. Yuan, A nonlinear conjugate gradient with a strong global convergence property, SIAM J. Opt. 10, 177 (1999).
  • D. D’Alessandro, Introduction to Quantum Control and Dynamics, Chapman & Hall (2008).
  • F. Dalfovo, G. Stefano, L.P. Pitaevskii, S. Stringari, Theory of Bose–Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999). [Crossref]
  • E. B. Davies, Quantum theory of open systems, Academic Press, London (1976).
  • I. Degani and A. Zanna, Optimal Quantum Control by an Adapted Coordinate Ascent Algorithm, SIAM J. Sci. Comput. 34, 1488 (2012).
  • I. Degani, A. Zanna, L. Saelen and R. Nepstad, Quantum Control With Piecewise Constant Control Functions, SIAM J. Sci. Comput. 31, 3566 (2009).
  • G. Dirr and U. Helmke, Lie Theory for Quantum Control, GAMM-Mitteilungen 31, 59 (2008). [Crossref]
  • L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, Rhode Island (2002).
  • H.O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications, vol. 62, Cambridge University Press (1999).
  • C. Fiolhais, F. Nogueira, and M. Marques (Eds.), A Primer in Density Functional Theory, Lectures Notes in Physics Vol. 620, Springer, Berlin (2003).
  • N Gisin and I C Percival, The quantum-state diffusion model applied to open systems, J. Phys. A: Math. Gen. 25, 5677 (1992). [Crossref]
  • V. Gorini, A. Kossakowski, and E.C.G. Sudarshan, Completely positive semigroups of N-level systems, J. Math. Phys. 17, 821 (1976). [Crossref]
  • W.W. Hager and H. Zhang, A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search, SIAM Journal on Optimization 16, 170 (2005). [Crossref]
  • U. Hohenester, Optical properties of semiconductor nanostructures: Decoherence versus quantum control, in M. Rieth and W. Schommers, (Eds.), Handbook of Theoretical and Computational Nanotechnology, American Scientific Publishers (2005).
  • U. Hohenester, P.-K. Rekdal, A. Borzì, J. Schmiedmayer, Optimal quantum control of Bose Einstein condensates in magnetic microtraps, Phys. Rev. A 75, 023602 (2007). [Crossref]
  • G. M. Huang, T. J. Tarn, and J. W. Clark, On the controllability of quantummechanical systems, J. Math. Phys. 24, 2608 (1983). [Crossref]
  • R. Illner, H. Lange, and H. Teismann, Limitations on the control of Schrödinger equations, ESAIM: COCV 12, 615 (2006).
  • W. Karwowski and R.V. Mendes, Quantum control in infinite dimensions, Phys. Lett. A 322, 282 (2004). [Crossref]
  • N. Khaneja, R. W. Brockett, and S. J. Glaser, Time Optimal Control in Spin Systems, Phys. Rev. A 63, 032308 (2001). [Crossref]
  • N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms, Journal of Magnetic Resonance 172, 296 (2005). [Crossref]
  • K. Kime, Simultaneous control of a rod equation and a simple Schrodinger equation, Systems and Control Letters 24, 301 (1995).
  • W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [Crossref]
  • K. Kormann, S. Holmgren and H. O. Karlsson, A Fourier-Coefficient Based Solution of an Optimal Control Problem in Quantum Chemistry, JOTA 147, 491 (2010).
  • I. Lesanovsky, T. Schumm, S. Hofferberth, L.M. Andersson, P. Krüger, and J. Schmiedmayer, Adiabatic radiofrequency potentials for the coherent manipulation of matter waves, Phys. Rev. A 73, 033619 (2006). [Crossref]
  • C. Lan, T.-J. Tarn, Q.-S. Chi, and J. W. Clark, Analytic controllability of time-dependent quantum control systems, J. Math. Phys. 46, 052102 (2005). [Crossref]
  • H. Lan and Y. Ding, Ordering, positioning and uniformity of quantum dot arrays, Nano Today 7, 94 (2012) . [Crossref]
  • P. Laurent, H. Rabitz, J. Salomon, and G. Turinici, Control through operators for quantum chemistry, Proceedings of the 51st IEEE Conference on Decision and Control, Maui (2012).
  • G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976). [Crossref]
  • J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin (1971).
  • J.L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review (1988).
  • Y. Maday, J. Salomon and G. Turinici, Monotonic time-discretized schemes in quantum control, Num. Math. 103, 323 (2006).
  • Y. Maday and G. Turinici, New formulations of monotonically convergent quantum control algorithms, J. Chem. Phys. 118, 8191 (2003). [Crossref]
  • Y. Maday, J. Salomon, and G. Turinici, Parareal in time control for quantum systems, SIAM J. Num. Anal. 45, 2468 (2007).
  • R. Melnik, Multiple scales and coupled effects in modelling low dimensional semiconductor nanostructures: Between atomistic and continuum worlds, Encyclopedia of Complexity and Systems Science, Editor-in-Chief Meyers, R.A., Springer (2009).
  • A.P. Peirce, M.A. Dahleh, and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications, Phys. Rev. A 37, 4950 (1988). [PubMed][Crossref]
  • H. Rabitz, G. Turinici, and E. Brown, Control of Molecular Motion: Concepts, Procedures, and Future Prospects, Ch. 14 in Handbook of Numerical Analysis, Volume X, P. Ciarlet and J. Lions, Eds., Elsevier, Amsterdam (2003).
  • H. Rabitz, M. Hsieh, and C. Rosenthal, Quantum optimally controlled transition landscapes, Science 303, 998 (2004).
  • V. Ramakrishna, H. Rabitz, M.V. Salapaka, M. Dahleh and A. Peirce, Controllability of Molecular Systems, Phys. Rev. A 62, 960 (1995). [Crossref]
  • E. Räsänen, A. Castro, J. Werschnik, A. Rubio, and E.K.U. Gross, Optimal Control of Quantum Rings by Terahertz Laser Pulses, Phys. Rev. Lett. 98, 157404 (2007). [PubMed][Crossref]
  • H. Risken, The Fokker Planck Equation: Methods of Solution and Applications, Springer.
  • Y. Saad, J.R. Chelikowsky, and S.M. Shontz, Numerical methods for electronic structure calculations of materials, SIAM Review 52, 3 (2010). [Crossref]
  • A. Sacchetti, Nonlinear time-dependent one-dimensional Schrodinger equation with double-well potential, SIAM J. Math. Anal. 35, 1160 (2004).
  • S.G. Schirmer, H. Fu, and A. Solomon, Complete controllability of quantum systems, Phys. Rev. A 63, 063410 (2001). [Crossref]
  • B.J. Sussman, D. Townsend, M.Y. Ivanov, and A. Stolow, Dynamic Stark control of photochemical processes, Science 13, 278 (2006). [Crossref]
  • G. Turinici, Controllable quantities for bilinear quantum systems, Proceedings of the 39th IEEE Conference on Decision and Control, 1364 (2000).
  • G. Turinici and H. Rabitz, Wavefunction controllability in quantum systems, Journal of Physics A 36, 2565 (2003).
  • A. van den Bos, Complex Gradient and Hessian, IEEE Proc.-Vis. Image Signal Processing 141, 380 (1994).
  • R. Vilela Mendes, Universal families and quantum control in infinite dimensions, Phys. Lett. A 373, 2529 (2009).
  • R. Vilela Mendes and V. I. Man’ko, Quantum control and the Strocchi map, Phys. Rev. A 67, 053404 (2003). [Crossref]
  • R. Vilela Mendes and V. I. Man’ko, On the problem of quantum control in infinite dimensions, J. Phys. A: Math. Theor. 44, 135302 (2011). [Crossref]
  • G. von Winckel and A. Borzì, Computational techniques for a quantum control problem with H1-cost, Inverse Problems 24, 034007 (2008).
  • G. von Winckel and A. Borzì, QUCON: A fast Krylov-Newton code for dipole quantum control problems, Computer Physics Communications, 181, 2158 (2010).
  • G. von Winckel, A. Borzì, and S. Volkwein, A globalized Newton method for the accurate solution of a dipole quantum control problem, SIAM J. Sci. Comput. 31, 4176 (2009).
  • H. M. Wiseman and G. J. Milburn, Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere, Phys. Rev. A 47, 1652 (1993). [PubMed][Crossref]
  • R. Wu, A. Pechen, C. Brif, and H. Rabitz, Controllability of open quantum systems with Kraus-map dynamics, J. Phys. A: Math. Theor. 40, 5681 (2007). [Crossref]
  • H. Yserentant, Regularity and Approximability of Electronic Wave Functions, Lecture Notes, Berlin, (2009).

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