A robust algorithm for finding phase factors in quantum signal processing*

Quantum Signal Processing (QSP) provides a general way to implement matrix functions on quantum computers. The algorithm can be efficiently used to solve quantum linear systems, to perform Hamiltonian simulation, and to prepare Gibbs ensembles, among other applications. QSP can exactly encode a degree-d polynomial transformation of a matrix using d+1 phase factors. However, the current strategies for solving for the phase factors of a given function can be numerically unstable. We present an efficient method to find the phase factors for a general real function, and demonstrate the performance for solving linear systems and eigenvalue problems.