We provide a complete solution to the problem of infinite quantum signal processing for the class of Szeg\H o functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a quantum signal processing representation. We do so by introducing a new algorithm called the Riemann-Hilbert-Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szeg\H o function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.

翻译：我们针对Szeg\H o函数类——即满足对数可积条件且几乎涵盖所有允许量子信号处理表示的函数——的无限量子信号处理问题提供了完整解决方案。为此，我们引入了一种称为Riemann-Hilbert-Weiss算法的新算法，该算法能够独立于所有其他相位因子计算任意单个相位因子。本算法同时也是首个被证明可稳定计算任意Szeg\H o函数相位因子的数值算法。稳定性证明涉及运用谱理论方法求解非线性傅里叶分析中的Riemann-Hilbert分解问题。