标题： 量子信号处理的数学和数值分析 显示英文标题

标题： Mathematical and numerical analysis of quantum signal processing

摘要： 量子信号处理（QSP）提供了一种表示次数为$d$的标量多项式的框架，该框架通过矩阵的乘积形式在$\mathrm{SU}(2)$中进行参数化，参数由$(d+1)$个称为相位因子的实数组成。 QSP 是量子奇异值变换（QSVT）的数学基础，后者通常被认为是过去十年最重要的量子算法之一，在科学计算中有广泛的应用，从哈密顿量模拟到求解线性方程组和特征值问题。 在本文中，我们综述了 QSP 在数学和数值分析方面的最新进展。 特别是，我们关注其超越多项式的一般化、相位因子评估算法的计算复杂性以及此类算法的数值稳定性。 解决这些问题中的某些问题依赖于 QSP、非线性傅里叶分析在$\mathrm{SU}(2)$上、快速多项式乘法以及具有位移结构的矩阵的高斯消去之间的意外相互作用。 显示英文摘要

摘要： Quantum signal processing (QSP) provides a representation of scalar polynomials of degree $d$ as products of matrices in $\mathrm{SU}(2)$, parameterized by $(d+1)$ real numbers known as phase factors. QSP is the mathematical foundation of quantum singular value transformation (QSVT), which is often regarded as one of the most important quantum algorithms of the past decade, with a wide range of applications in scientific computing, from Hamiltonian simulation to solving linear systems of equations and eigenvalue problems. In this article we survey recent advances in the mathematical and numerical analysis of QSP. In particular, we focus on its generalization beyond polynomials, the computational complexity of algorithms for phase factor evaluation, and the numerical stability of such algorithms. The resolution to some of these problems relies on an unexpected interplay between QSP, nonlinear Fourier analysis on $\mathrm{SU}(2)$, fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.