# On variants of multivariate quantum signal processing and their characterizations

arXiv: https://arxiv.org/abs/2312.09072 · HTML: https://mlsanity.com/paper/2312.09072/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2023.

## Built on by (most cited first, 1 of 1)

- [Expressive and Scalable Quantum Fusion for Multimodal Learning](https://mlsanity.com/paper/2510.06938/) (2025)

## Abstract (arXiv)

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.