For a right-invariant and controllable driftless system on SU(n), we consider a time-periodic reference trajectory along which the linearized control system generates su(n): such trajectories always exist and constitute the basic ingredient of Coron's Return Method. The open-loop controls that we propose, which rely on a left-invariant tracking error dynamics and on a fidelity-like Lyapunov function, are determined from a finite number of left-translations of the tracking error and they assure global asymptotic convergence towards the periodic reference trajectory. The role of these translations is to avoid being trapped in the critical region of this Lyapunov-like function. The convergence proof relies on a periodic version of LaSalle's invariance principle and the control values are determined by numerical integration of the dynamics of the system. Simulations illustrate the obtained controls for n = 4 and the generation of the C-NOT quantum gate.

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This paper considers right-invariant and controllable driftless quantum systems with state X(t) evolving on the unitary group U(n) and m inputs u = (u 1 , . . . , u m ). The T -sampling stabilisation problem is introduced and solved: given any initial condition X 0 and any goal state X goal , find a control law u = u(X, t) such that lim j →∞ X(jT ) = X goal for the closed-loop system. The purpose is to generate arbitrary quantum gates corresponding to X goal . This is achieved by the tracking of T -periodic reference trajectories (X a (t), u a (t)) of the quantum system that pass by X goal using the framework of Coron's return method. The T -periodic reference trajectories X a (t) are generated by applying controls u a (t) that are a sum of a finite number M of harmonics of sin(2πt/T ), whose amplitudes are parameterised by a vector a. The main result establishes that, for M big enough, X(jT ) exponentially converges towards X goal for almost all fixed a, with explicit and completely constructive control laws. This paper also establishes a stochastic version of this deterministic control law. The key idea is to randomly choose a different parameter vector of control amplitudes a = a j at each t = jT , and keeping it fixed for t ∈ [jT , (j + 1)T ). It is shown in the paper that X(jT ) exponentially converges towards X goal almost surely. Simulation results have indicated that the convergence speed of X(jT ) may be significantly improved with such stochastic technique. This is illustrated in the generation of the C-NOT quantum logic gate on U(4).

Minha mais profunda gratidão aos meus orientadores, Prof. Dr. Paulo Sérgio Pereira da Silva e Prof. Dr. Pierre Rouchon, por me aceitarem como aluno, por estarem constantemente presentes, por me guiarem com paciência e segurança, pela generosidade em ensinar e em remover minhas dúvidas, pelas incontáveis vezes em que passamos horas e horas conversando, discutindo e preenchendo todo o quadro com esboços de novas idéias (os rabiscos no quadro pareciam uma obra de arte!), pelo comprometimento em realizar estudos do mais alto nível acadêmico, pela gentileza e amizade, e por sempre me mostrarem o quanto ainda tenho que aprender quando eu, erroneamente, acho que sei alguma coisa. Sinto-me privilegiado por ser aluno de mestres como vocês. Muito obrigado! A CAPES, pelo financiamento integral da presente tese e pela oportunidade em realizar um estágio de doutorado na França sob a orientação do Prof. Dr. Pierre Rouchon.

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