Quantum signal processing (QSP) is a technique that was shown to unify and simplify many new and known quantum algorithms. Its powerfulness lies in the possibility of carrying out eigenvalue or singular value transformations of block-encoded matrices. Recent works have found a connection between QSP and non-linear Fourier analysis, showing that a QSP protocol for a desired transformation can be stably computed by inverting the so-called non-linear Fourier transform (NLFT). In this work we strengthen the connection between NLFT and QSP, by showing that the NLFT over general complex sequences can be turned into a generalized QSP (GQSP) protocol [Motlagh, Wiebe ’24] and viceversa. In other words, the full theory of single-qubit QSP and the theory of NLFT over SU(2) are the same, giving the former a solid mathematical foundation. In this talk I will introduce both (G)QSP and NLFT and show how QSP benefits from this insight.

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