Generalized Quantum Signal Processing and Non-Linear Fourier Transform are equivalent

• URL: http://arxiv.org/abs/2503.03026v1
• Date: Tue, 04 Mar 2025 22:02:38 GMT
• Title: Generalized Quantum Signal Processing and Non-Linear Fourier Transform are equivalent
• Authors: Lorenzo Laneve,
• Abstract summary: Quantum signal processing (QSP) and quantum singular value transformation (QSVT) are powerful techniques for the development of quantum procedures.<n>Recent research showed that Non-Linear Fourier Analysis (NLFA) can be employed to numerically compute a QSP protocol, with provable stability.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum signal processing (QSP) and quantum singular value transformation (QSVT) are powerful techniques for the development of quantum procedures. They allow to derive circuits preparing desired polynomial transformations. Recent research [Alexis et al. 2024] showed that Non-Linear Fourier Analysis (NLFA) can be employed to numerically compute a QSP protocol, with provable stability. In this work we extend their result, showing that GQSP and the Non-Linear Fourier Transform over $SU(2)$ are the same object. This statement - proven by a simple argument - has a bunch of consequences: first, the Riemann-Hilbert-Weiss algorithm can be turned, with little modifications and no penalty in complexity, into a unified, provably stable algorithm for the computation of phase factors in any QSP variant, including GQSP. Secondly, we derive a uniqueness result for the existence of GQSP phase factors based on the bijectivity of the Non-Linear Fourier Transform. Furthermore, NLFA provides a complete theory of infinite generalized quantum signal processing, which characterizes the class of functions approximable by GQSP protocols.

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