Polynomial time constructive decision algorithm for multivariable quantum signal processing

TL;DR

This paper introduces a polynomial-time classical algorithm to determine if a pair of multivariable Laurent polynomials can be implemented by multivariable quantum signal processing, providing a constructive method for parameter selection.

Contribution

It presents the first efficient algorithm with necessary and sufficient conditions for implementing multivariable quantum signal processing.

Findings

The algorithm runs in polynomial time relative to variables and operators.

It provides a constructive method for parameter selection in M-QSP.

It establishes necessary and sufficient conditions for polynomial implementability.

Abstract

Quantum signal processing (QSP) and quantum singular value transformation (QSVT) have provided a unified framework for understanding many quantum algorithms, including factorization, matrix inversion, and Hamiltonian simulation. As a multivariable version of QSP, multivariable quantum signal processing (M-QSP) is proposed. M-QSP interleaves signal operators corresponding to each variable with signal processing operators, which provides an efficient means to perform multivariable polynomial transformations. However, the necessary and sufficient condition for what types of polynomials can be constructed by M-QSP is unknown. In this paper, we propose a classical algorithm to determine whether a given pair of multivariable Laurent polynomials can be implemented by M-QSP, which returns True or False. As one of the most important properties of this algorithm, its returning True is the necessary and sufficient condition. The proposed classical algorithm runs in polynomial time in the number of variables and signal operators. Our algorithm also provides a constructive method to select the necessary parameters for implementing M-QSP. These findings offer valuable insights for identifying practical applications of M-QSP.