Efficient phase-factor evaluation in quantum signal processing

Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on computers. Asymptotic analysis of algorithms based QSP has shown that asymptotically optimal results can in principle be obtained for range tasks, such as Hamiltonian simulation and the linear system problem. A further benefit it uses minimal number ancilla qubits, which facilitates its implementation near-to-intermediate term architectures. However, there so far no classically stable allowing computation phase factors are needed build circuits. Existing methods require use variable precision arithmetic only applied relatively low degree. We present here an optimization-based method accurately compute using standard double operations. demonstrate performance this approach with applications simulation, eigenvalue filtering, problems. Our numerical show optimization find approximate degree larger than $10\phantom{\rule{0.16em}{0ex}}000$ errors below ${10}^{\ensuremath{-}12}$.