On variants of multivariate quantum signal processing and their characterizations

• URL: http://arxiv.org/abs/2312.09072v1
• Date: Thu, 14 Dec 2023 16:06:58 GMT
• Title: On variants of multivariate quantum signal processing and their characterizations
• Authors: Bal\'azs N\'emeth, Blanka K\"ov\'er, Bogl\'arka Kulcs\'ar, Roland Botond Mikl\'osi, Andr\'as Gily\'en
• Abstract summary: Quantum signal processing (QSP) is a highly successful algorithmic primitive in quantum computing. We show that Haah's characterization of general QSP can be extended to homogeneous bivariable (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.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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