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Optimized single-qubit gates for Josephson phase qubits

In a Josephson phase qubit the coherent manipulations of the computational states are achieved by modulating an applied ac current, typically in the microwave range. In this work we show that it is possible to find optimal modulations of the bias current to achieve high-fidelity gates. We apply quantum optimal control theory to determine the form of the pulses and study in details the case of a NOT-gate. To test the efficiency of the optimized pulses in an experimental setup, we also address the effect of possible imperfections in the pulses shapes, the role of off-resonance elements in the Hamiltonian, and the effect of capacitive interaction with a second qubit.