Product decomposition of periodic functions in quantum signal processing

Abstract

We consider an algorithm to approximate complex-valued periodic functions f(eiθ) as a matrix element of a product of SU(2)-valued functions, which underlies so-called quantum signal processing. We prove that the algorithm runs in time O(N3polylog(N/ε)) under the random-access memory model of computation where N is the degree of the polynomial that approximates f with accuracy ε; previous efficiency claim assumed a strong arithmetic model of computation and lacked numerical stability analysis.