  This paper addresses the problem of solving nonlinear systems in the context
of symmetric quantum signal processing (QSP), a powerful technique for
implementing matrix functions on quantum computers. Symmetric QSP focuses on
representing target polynomials as products of matrices in SU(2) that possess
symmetry properties. We present a novel Newton's method tailored for
efficiently solving the nonlinear system involved in determining the phase
factors within the symmetric QSP framework. Our method demonstrates rapid and
robust convergence in all parameter regimes, including the challenging scenario
with ill-conditioned Jacobian matrices, using standard double precision
arithmetic operations. For instance, solving symmetric QSP for a highly
oscillatory target function $\alpha \cos(1000 x)$ (polynomial degree $\approx
1433$) takes $6$ iterations to converge to machine precision when $\alpha=0.9$,
and the number of iterations only increases to $18$ iterations when
$\alpha=1-10^{-9}$ with a highly ill-conditioned Jacobian matrix. Leveraging
the matrix product states the structure of symmetric QSP, the computation of
the Jacobian matrix incurs a computational cost comparable to a single function
evaluation. Moreover, we introduce a reformulation of symmetric QSP using
real-number arithmetics, further enhancing the method's efficiency. Extensive
numerical tests validate the effectiveness and robustness of our approach,
which has been implemented in the QSPPACK software package.
