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arXiv · 2312.09072

On variants of multivariate quantum signal processing and their characterizations

Balázs Németh·Blanka Kövér·Boglárka Kulcsár·Roland Botond Miklósi·András Gilyén
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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.

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Balázs Németh, Blanka Kövér, Boglárka Kulcsár, Roland Botond Miklósi, András Gilyén. 2023-12-14. On variants of multivariate quantum signal processing and their characterizations. https://arxiv.org/abs/2312.09072

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