The Hitchhiker's Guide to QSP pre-processing

TL;DR

This paper reviews quantum signal processing pre-processing techniques, compares different methods, and introduces benchmarks showing the superior performance of a recent method over existing approaches.

Contribution

It provides a comprehensive, accessible review of QSP pre-processing conventions, compares multiple methods, and benchmarks a new Wilson method against existing strategies.

Findings

Wilson method succeeds with linear error propagation in relevant regimes

Berntson and S"underhauf's method outperforms Wilson and standard optimization

Benchmarking shows the new method is more efficient for complex polynomials

Abstract

Quantum signal processing (QSP) relies on a historically costly pre-processing step, "QSP-processing/phase-factor finding." QSP-processing is now a developed topic within quantum algorithms literature, and a beginner accessible review of QSP-processing is overdue. This work provides a whirlwind tour through QSP conventions and pre-processing methods, beginning from a pedagogically accessible QSP convention. We then review QSP conventions associated with three common polynomial types: real polynomials with definite parity, sums of reciprocal/anti-reciprocal Chebyshev polynomials, and complex polynomials. We demonstrate how the conventions perform with respect to three criteria: circuit length, polynomial conditions, and pre-processing methods. We then review the recently introduced Wilson method for QSP-processing and give conditions where it can succeed with bound error. Although the resulting bound is not computationally efficient, we demonstrate that the method succeeds with linear error propagation for relevant target polynomials and precision regimes, including the Jacobi-Anger expansion used in Hamiltonian simulation algorithms. We then apply our benchmarks to three QSP-processing methods for QSP circuits and show that a method introduced by Berntson and S\"underhauf outperforms both the Wilson method and the standard optimization strategy for complex polynomials.