Nonlinear transformation of complex amplitudes via quantum singular value transformation - Naixu Guo; Kosuke Mitarai; Keisuke Fujii (Physical Review Research, 2024)
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 achieve. In this paper, 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. In this paper, we provide a promising way to introduce the highly complex nonlinearity of the quantum states, which is essentially missing in quantum mechanics.