On multivariate polynomials achievable with quantum signal processing

TL;DR

This paper investigates which multivariate polynomials can be constructed using quantum signal processing, providing new necessary and sufficient conditions for their decomposability, expanding the understanding of QSP's capabilities.

Contribution

It introduces a new formalism and proves the first sufficient condition for constructing multivariate polynomials with QSP.

Findings

Derived simpler necessary conditions for polynomial decomposability.

Established the first proven sufficient condition for multivariate polynomial construction in QSP.

Enhanced understanding of the limits and possibilities of quantum signal processing.

Abstract

Quantum signal processing (QSP) is a framework which was proven to unify and simplify a large number of known quantum algorithms, as well as discovering new ones. QSP allows one to transform a signal embedded in a given unitary using polynomials. Characterizing which polynomials can be achieved with QSP protocols is an important part of the power of this technique, and while such a characterization is well-understood in the case of univariate signals, it is unclear which multivariate polynomials can be constructed when the signal is a vector, rather than a scalar. This work uses a slightly different formalism than what is found in the literature, and uses it to find simpler necessary conditions for decomposability, as well as a sufficient condition -- the first, to the best of our knowledge, proven for a (generally inhomogeneous) multivariate polynomial in the context of quantum signal processing.