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.