Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. It represents a real scalar polynomial of degree d using a product of unitary matrices of size 2*2, parameterized by (d+1) real numbers called the phase factors. For a given function f, phase factors can be obtained by solving an optimization problem. However, the cost function is non-convex, and has a very complex energy landscape with numerous global and local minima. We provides a partial explanation of the success of our optimization based algorithm. Then we propose another simple iterative algorithm, fixed point iteration, which is inspired by our theoretical analysis. This is the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit d ? 8. We also present a novel Newton's method, which demonstrates rapid and robust convergence in all parameter regimes, including the challenging scenario with ill-conditioned Jacobian matrices. Extensive numerical tests validate the effectiveness and robustness of our approaches, which has been implemented in the QSPPACK software package.
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