Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916745035382784 |
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| author | Ni, Hongkang Sarkar, Rahul Ying, Lexing Lin, Lin |
| author_facet | Ni, Hongkang Sarkar, Rahul Ying, Lexing Lin, Lin |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing Ni, Hongkang Sarkar, Rahul Ying, Lexing Lin, Lin Quantum Physics Numerical Analysis 65T50, 68W40 (Primary) 65Y20, 68Q12, 81P68, 43A50 (Secondary) 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). |
| title | Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing |
| topic | Quantum Physics Numerical Analysis 65T50, 68W40 (Primary) 65Y20, 68Q12, 81P68, 43A50 (Secondary) |
| url | https://arxiv.org/abs/2505.12615 |