Complex numbers are necessary for the continuous time evolution of quantum gates, since real special orthogonal operators cannot represent gates with determinant -1 and real-to-complex mappings remain isomorphic to complex representations.
Real- vector-space quantum theory with a universal quantum bit.Physical Review A—Atomic, Molecular, and Optical Physics, 87(5):052106, 2013
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Time evolution of quantum gates and the necessity of complex numbers
Complex numbers are necessary for the continuous time evolution of quantum gates, since real special orthogonal operators cannot represent gates with determinant -1 and real-to-complex mappings remain isomorphic to complex representations.