An efficient computational method for the optimal control of higher dimensional quantum systems

The process of obtaining solutions to optimal control problems via mesh based techniques suffers from the well known problem of the curse of dimensionality. This issue is especially severe in quantum systems whose dimensions grow exponentially with the number of interacting elements (qubits) that they contain. In this article we develop a modification of recently introduced curse-of-dimensionality-free max-plus techniques, to the control of quantum systems. This method yields a more manageable growth that is related to the cardinality of the control set. Its efficacy is demonstrated by obtaining an approximate solution to a previously intractable problem on a 2 qubit system.

[1]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[2]  Aharon Ben-Tal,et al.  Lectures on modern convex optimization , 1987 .

[3]  Mile Gu,et al.  Quantum Computation as Geometry , 2006, Science.

[4]  William M. McEneaney,et al.  Max-plus methods for nonlinear control and estimation , 2005 .

[5]  Anthony M. Bloch,et al.  Nonlinear Dynamical Control Systems (H. Nijmeijer and A. J. van der Schaft) , 1991, SIAM Review.

[6]  Uwe Helmke,et al.  Spin Dynamics: A Paradigm for Time Optimal Control on Compact Lie Groups , 2006, J. Glob. Optim..

[7]  B. Hall.  Lie Groups, Lie Algebras, and Representations: An Elementary Introduction , 2004 .

[8]  M. James,et al.  Gate complexity using dynamic programming , 2008, 0807.0507.

[9]  A. Doherty,et al.  Optimal control, geometry, and quantum computing , 2006, quant-ph/0603160.

[10]  Navin Khaneja,et al.  Cartan decomposition of SU(2n) and control of spin systems , 2001 .

[11]  Mark R. Sepanski,et al.  Compact Lie Groups , 2006 .

[12]  R. Brockett,et al.  Time optimal control in spin systems , 2000, quant-ph/0006114.

[13]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[14]  William M. McEneaney,et al.  A Curse-of-Dimensionality-Free Numerical Method for Solution of Certain HJB PDEs , 2007, SIAM J. Control. Optim..

[15]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[16]  W.M. McEneaney,et al.  Curse-of-complexity attenuation in the curse-of-dimensionality-free method for HJB PDEs , 2008, 2008 American Control Conference.

[17]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.