Amplitude Transformation of Quantum State Based on QSVT

Quantum singular value transformation (QSVT) is a powerful quantum algorithm that can effectively perform a polynomial transformation of the singular values of a linear operator embedded in a unitary matrix. This article extends QSVT to perform a polynomial transformation of the amplitudes of a quantum state. This article presents an algorithm that can convert quantum state $\sum_{i=0}^{N-1}x_{i}\vert i\rangle$ to quantum state $\frac{1}{\sqrt{\sum_{i=0}^{N-1}\vert f(x_{i})\vert^{2}}}\sum_{i=0}^{N-1}f(x_{i})\vert\rangle$ and demonstrates the application of the algorithm in the quantum state small amplitude extraction.