# Nonlinear transformation of complex amplitudes via quantum singular value transformation

arXiv: https://arxiv.org/abs/2107.10764 · HTML: https://mlsanity.com/paper/2107.10764/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2021.

## Built on by (most cited first, 9 of 9)

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- [Post-variational quantum neural networks](https://mlsanity.com/paper/2307.10560/) (2023)
- [Accelerating Inference for Multilayer Neural Networks with Quantum Computers](https://mlsanity.com/paper/2510.07195/) (2025)
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- [Statistics-Informed Parameterized Quantum Circuit via Maximum Entropy Principle for Data Science and Finance](https://mlsanity.com/paper/2406.01335/) (2024)

## Abstract (arXiv)

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.