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This paper addresses the problem of solving nonlinear systems in the contextof symmetric quantum signal processing (QSP), a powerful technique forimplementing matrix functions on quantum computers. Symmetric QSP focuses onrepresenting target polynomials as products of matrices in SU(2) that possesssymmetry properties. We present a novel Newton's method tailored forefficiently solving the nonlinear system involved in determining the phasefactors within the symmetric QSP framework. Our method demonstrates rapid androbust convergence in all parameter regimes, including the challenging scenariowith ill-conditioned Jacobian matrices, using standard double precisionarithmetic operations. For instance, solving symmetric QSP for a highlyoscillatory target function $\alpha \cos(1000 x)$ (polynomial degree $\approx1433$) 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. Leveragingthe matrix product states the structure of symmetric QSP, the computation ofthe Jacobian matrix incurs a computational cost comparable to a single functionevaluation. Moreover, we introduce a reformulation of symmetric QSP usingreal-number arithmetics, further enhancing the method's efficiency. Extensivenumerical tests validate the effectiveness and robustness of our approach,which has been implemented in the QSPPACK software package.

Robust iterative method for symmetric quantum signal processing in all parameter regimes