  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.
