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International Journal of Theoretical Physics

Quantum Windowed Fourier Transform and its Application to Quantum Signal Processing

Haiting Yin, Dayong Lu, Rui Zhang

Published Sep 15, 2021• 12 citations
Publisher
DOI
10.1007/s10773-021-04933-7

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The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on SU(1,1)\mathrm{SU}(1,1)SU(1,1) has been widely studied, its SU(2)\mathrm{SU}(2)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 SU(2)\mathrm{SU}(2)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).
We define a nonlinear Fourier transform which maps sequences of contractive n×nn \times nn×n matrices to SU(2n)SU(2n)SU(2n)-valued functions on the circle T\mathbb{T}T. We characterize the image of finitely supported sequences and square-summable sequences on the half-line, and construct an inverse for SU(2n)SU(2n)SU(2n)-valued functions whose diagonal n×nn \times nn×n blocks are outer matrix functions. As an application, we relate this nonlinear Fourier transform with quantum signal processing over U(2n)U(2n)U(2n) and multivariate quantum signal processing.