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In this paper, we study the manipulation of a two level quantum mechanical system which is the basic unit of representing data in quantum information science, a qubit. This research is motivated by the design of the quantum Hadamard gate for solid state spin qubits. We consider the problem of driving the initial state of a qubit into a desired final state in a specified time, while minimizing the energies of the control fields. Optimal control theory is employed to define optimum control fields which satisfy both dynamical constraints and optimality conditions. Time evolution of the system is calculated within the Schrödinger equation and a gradient method is used for tailoring optimal control fields.