Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing

• URL: http://arxiv.org/abs/2505.12615v1
• Date: Mon, 19 May 2025 01:57:04 GMT
• Title: Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing
• Abstract summary: We investigate the inverse NLFT on $mathrmSU(2)$ and establish the numerical stability of the layer stripping algorithm for the first time.<n>We develop a fast and numerically stable algorithm, called inverse nonlinear fast Fourier transform, for performing inverse NLFT with near-linear complexity.
• Score: 10.193028145901248
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on $\mathrm{SU}(1,1)$ has been widely studied, its $\mathrm{SU}(2)$ variant has only recently attracted attention due to emerging applications in quantum signal processing (QSP) and quantum singular value transformation (QSVT). In this paper, we investigate the inverse NLFT on $\mathrm{SU}(2)$ and establish the numerical stability of the layer stripping algorithm for the first time under suitable conditions. Furthermore, we develop a fast and numerically stable algorithm, called inverse nonlinear fast Fourier transform, for performing inverse NLFT with near-linear complexity. This algorithm is applicable to computing phase factors for both QSP and the generalized QSP (GQSP).

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