Control of Quantum Gates and of Ensembles of Bloch Equations

Abstract

The basic ingredients of this research plan is the nonlinear control theory, which will be applied for searching the solution of quantum control problems. Due to the structureof the models that are associated to the Schrodinger equation, these control problems are suitable for the application of differential-geometric methods, since the state space is the Lie-Group U(n) of unitary matrices (or SU(n), if the global phase is disregarded). The first problem to be studied is the synthesis of quantum gates, which are the building blocks of the future quantum computer.The idea is the optimization of previous results that were obtained by the researcher, where Lyapunov-LaSalle stabilization techniques were applied for obtaining the quantum gate generation. When one says optimization here, one does not mean the application of standard optimal control techniques, due to the known complexity barriers that are associated to optimal control for quantum systems. Indeed, one will consider iterative techniques. One will develop new iterative algorithms that are implementable in the existent computational machines. The second problem to be considered is the control of Ensembles of Bloch equations, which is a fundamental problem that appears in the context of magnetic resonance. In previous works, we have already developed Lyapunov techniques for solving this problem, but uses a comb of Dirac pulses (spin-echo pulses). The new idea is the replacement of these Dirac pulses by an adiabatic control. This will allow the construction of small amplitude pulses that will have the same effect of the spin-echo pulses. (AU)