MINES ParisTech CAS - Centre automatique et systèmes


Séance du jeudi 4 juillet 2013, salle L106, 14h.
Paulo Sergio PEREIRA DA SILVA, Associate Professor, Laboratorio de Automação e Controle, Escola Politécnica, Universidade de São Paulo, Brazil

This work considers the problem of generating any desired goal propagator for a driftless quantum system that evolves on the special unitary group SU(n) (or U(n)) . The relevance of such control problem is, for instance, the realization of arbitrary quantum gates in quantum computers. Assuming only the controllability of the system, we construct explicit control laws that assure global asymptotic convergence of the propagator of the system towards the goal propagator. The controls laws rely on a reference trajectory that crosses the desired goal propagator in a time-periodic fashion and is such that its corresponding linearised system generates the Lie algebra su(n) (or u(n)). Their existence is ensured by the Return Method of Coron. The stability of the tracking of the reference trajectory is assured by a convenient application of a version LaSalle's Invariance Theorem [1]. The efficiency of the control law can be improved when one combines this control law with a random selection of some parameters that define the reference trajectory. In this case, closed loop stability can be shown (almost surely) as an application of Kushner's Theorem (a stochastic version of Lasalle's Invariance Theorem) [2]

[1] H. Bessa Silveira, P. S. Pereira da Silva, an P. Rouchon. Explicit Control Laws for the Periodic Motion Planning of Controllable Driftless Systems on SU(n). IEEE Conference on Decision and Control CDC'2012. p. 3640-3645.

[2] H. Bessa Silveira, P. S. Pereira da Silva, an P. Rouchon. A Stochastic Lyapunov Feedback Technique for Propagator Generation of Quantum Systems on U(n). Submitted to ECC'2013.