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Quantum interferometry and quantum information processing have been proposed for Bose-Einstein condensates (BECs), but BECs are described in complicated ways such as using quantum field theory or using a nonlinear differential equation. Nonlinear quantum mechanics does not mesh well with the superposition principle at the heart of interferometry and quantum information processing but could be compatible. Thus, we develop a rigorous foundation for quantum gates, obtained by solving the equation for evolution, and then we employ this foundation, combined with quantum-control techniques and appropriate state-sampling techniques, to devise feasible nonlinear Hadamard gates and thereby feasible, i.e., high-contrast, nonlinear Ramsey interferometry. Our approach to BEC interferometry and quantum logic shifts the paradigm by enlarging to the case of nonlinear quantum mechanics, which we apply to the cases of BEC interferometry and quantum information processing.