Product Decomposition of Periodic Functions in Quantum Signal Processing

TL;DR

This paper presents an efficient, numerically stable algorithm for approximating complex periodic functions using quantum signal processing, with proven polynomial-time complexity under realistic computational models.

Contribution

It introduces a new algorithm for quantum signal processing that is both efficient and numerically stable, improving upon previous methods.

Findings

Algorithm runs in O(N^3 polylog(N/ε)) time

Provides numerical stability analysis

Works under realistic RAM computational model

Abstract

We consider an algorithm to approximate complex-valued periodic functions $f(e^{i\theta})$ 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 $\mathcal O(N^3 \mathrm{polylog}(N/\epsilon))$ under the random-access memory model of computation where $N$ is the degree of the polynomial that approximates $f$ with accuracy $\epsilon$; previous efficiency claim assumed a strong arithmetic model of computation and lacked numerical stability analysis.