On variants of multivariate quantum signal processing and their characterizations

TL;DR

This paper extends the understanding of multivariate quantum signal processing, providing new characterizations for certain cases and highlighting limitations in existing conjectures, which could enable new quantum algorithms.

Contribution

It generalizes Haah's univariate QSP characterization to bivariate cases and introduces novel multivariate QSP variants beyond previous limitations.

Findings

Extended Haah's characterization to homogeneous bivariate QSP.

Refuted Rossi and Chuang's conjecture for certain multivariate cases.

Proposed new multivariate QSP variants that could lead to novel algorithms.

Abstract

Quantum signal processing (QSP) is a highly successful algorithmic primitive in quantum computing which leads to conceptually simple and efficient quantum algorithms using the block-encoding framework of quantum linear algebra. Multivariate variants of quantum signal processing (MQSP) could be a valuable tool in extending earlier results via implementing multivariate (matrix) polynomials. However, MQSP remains much less understood than its single-variate version lacking a clear characterization of "achievable" multivariate polynomials. We show that Haah's characterization of general univariate QSP can be extended to homogeneous bivariate (commuting) quantum signal processing. We also show a similar result for an alternative inhomogeneous variant when the degree in one of the variables is at most 1, but construct a counterexample where both variables have degree 2, which in turn refutes an earlier characterization proposed / conjectured by Rossi and Chuang for a related restricted class of MQSP. Finally, we describe homogeneous multivariate (non-commuting) QSP variants that break away from the earlier two-dimensional treatment limited by its reliance on Jordan-like decompositions, and might ultimately lead to the development of novel quantum algorithms.