Mostly Harmless Methods for QSP-Processing with Laurent Polynomials

TL;DR

This paper introduces a novel method for QSP-processing with Laurent polynomials that avoids optimization and root-finding, enabling efficient quantum algorithm development, especially for Hamiltonian simulation.

Contribution

The authors present a new QSP-processing technique for complex polynomials that does not rely on optimization or root-finding, improving efficiency and robustness.

Findings

Successfully applied to polynomials with floating point coefficients

Effective for Jacobi-Anger expansion in Hamiltonian simulation

Identifies regimes where existing methods struggle without arbitrary precision

Abstract

Quantum signal processing (QSP) and its extensions are increasingly popular frameworks for developing quantum algorithms. Yet QSP implementations still struggle to complete a classical pre-processing step ('QSP-processing') that determines the set of $SU(2)$ rotation matrices defining the QSP circuit. We introduce a method of QSP-processing for complex polynomials that identifies a solution without optimization or root-finding and verify the success of our methods with polynomials characterized by floating point precision coefficients. We demonstrate the success of our technique for relevant target polynomials and precision regimes, including the Jacobi-Anger expansion used in QSP Hamiltonian Simulation. For popular choices of sign and inverse function approximations, we characterize regimes where all known QSP-processing methods should be expected to struggle without arbitrary precision arithmetic.