Finding Angles for Quantum Signal Processing with Machine Precision

We describe an algorithm for finding angle sequences in quantum signal processing, with a novel component we call {\em halving} based on a new algebraic uniqueness theorem, and another we call capitalization. We present both theoretical and experimental results that demonstrate the performance of the new algorithm. In particular, these two algorithmic ideas allow us to find sequences of more than 3000 angles within 5 minutes for important applications such as Hamiltonian simulation, all in standard double precision arithmetic. This is native to almost all hardware.