Quantum Singular Value Transformation (QSVT) is one of the most important developments in quantum algorithms in the past decade. At the heart of QSVT is an innovative polynomial representation called Quantum Signal Processing (QSP), which can encode a target polynomial of definite parity using the product of a sequence of parameterized SU(2) matrices. Given a target polynomial, the corresponding parameters are called phase factors. In the past few years, there has been significant progress in designing and analyzing algorithms for finding phase factors, which can be viewed as a highly nonlinear optimization problem. In this talk, we argue that nonlinear Fourier analysis (NLFA) provides a natural framework for understanding QSP, as first observed by Thiele et al. Based on NLFA, we develop a Riemann--Hilbert--Weiss (RHW) algorithm to evaluate phase factors. To the best of our knowledge, this is the first provably numerically stable algorithm for almost all functions that admit a QSP representation. We will also discuss the impact of QSP on NLFA, which may lead to surprising progress in algorithms for inverse nonlinear Fourier transformations. https://arxiv.org/abs/2407.05634
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