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.

翻译：本文针对对称量子信号处理（QSP）中的非线性系统求解问题展开研究。对称量子信号处理是一种在量子计算机上实现矩阵函数的重要技术，其核心在于将目标多项式表示为具有对称性的SU(2)矩阵乘积。我们提出了一种针对对称QSP相位因子非线性系统求解的新型牛顿法，该方法在所有参数范围内（包括雅可比矩阵病态的挑战性场景）均展现出快速且鲁棒的收敛性能，且仅需标准双精度浮点运算。例如，当求解高度振荡目标函数$\alpha \cos(1000 x)$（多项式次数约1433次）的对称QSP问题时，在$\alpha=0.9$条件下仅需6次迭代即可收敛至机器精度；而当$\alpha=1-10^{-9}$且雅可比矩阵高度病态时，迭代次数仅增加至18次。利用对称QSP的矩阵乘积态结构，雅可比矩阵的计算代价与单次函数求值相当。此外，我们引入了基于实数算术的对称QSP重构方法，进一步提升了算法效率。大量数值实验验证了该方法的有效性和鲁棒性，相关算法已在QSPPACK软件包中实现。