  Due to the linearity of quantum operations, it is not straightforward to
implement nonlinear transformations on a quantum computer, making some
practical tasks like a neural network hard to be achieved. In this work, we
define a task called nonlinear transformation of complex amplitudes and provide
an algorithm to achieve this task. Specifically, we construct a block-encoding
of complex amplitudes from a state preparation unitary. This allows us to
transform the complex amplitudes by using quantum singular value
transformation. We evaluate the required overhead in terms of input dimension
and precision, which reveals that the algorithm depends on the roughly square
root of input dimension and achieves an exponential speedup on precision
compared with previous work. We also discuss its possible applications to
quantum machine learning, where complex amplitudes encoding classical or
quantum data are processed by the proposed method. This paper provides a
promising way to introduce highly complex nonlinearity of the quantum states,
which is essentially missing in quantum mechanics.
