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Due to the linearity of quantum operations, it is not straightforward toimplement nonlinear transformations on a quantum computer, making somepractical tasks like a neural network hard to be achieved. In this work, wedefine a task called nonlinear transformation of complex amplitudes and providean algorithm to achieve this task. Specifically, we construct a block-encodingof complex amplitudes from a state preparation oracle. This allows us totransform the complex amplitudes by using quantum singular valuetransformation. We evaluate the required overhead in terms of input dimensionand precision, which reveals that the algorithm depends on the roughly squareroot of input dimension and achieves an exponential speedup on precisioncompared with previous work. We also discuss its possible applications toquantum machine learning, where complex amplitudes encoding classical orquantum data are processed by the proposed method. This paper provides apromising way to introduce highly complex nonlinearity of the quantum states,which is essentially missing in quantum mechanics.

Nonlinear transformation of complex amplitudes via quantum singular value transformation