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.

翻译：由于量子操作的线性特性，在量子计算机上实现非线性变换并非易事，这使得神经网络等实际任务难以实现。本研究定义了一项名为复振幅非线性变换的任务，并给出实现该任务的算法。具体而言，我们通过态制备酉算子构建复振幅的分块编码，从而利用量子奇异值变换对复振幅进行变换。我们评估了该算法在输入维度和精度方面所需的开销，结果表明该算法对输入维度的依赖大致呈平方根关系，且相比于现有工作在精度上实现了指数级加速。我们还讨论了该算法在量子机器学习中的潜在应用——通过所提方法处理编码经典或量子数据的复振幅。本文为引入量子态中本质缺失的高度复杂非线性提供了有前景的途径。