Nonlinear Fourier analysis and connections to quantum computing

This lecture offers an introduction to the Nonlinear Fourier Transform (NLFT), a powerful analytic tool that extends classical Fourier analysis to nonlinear settings. We present the definition of the SU(2)-valued NLFT and explore its functorial properties with an emphasis on symmetries. We then focus on the nonlinear Plancherel theorem, which serves as a nonlinear analogue for the classical Plancherel identity. We also discuss a deep and elegant equivalence: up to a change of variables, the NLFT coincides with a well-known algorithm in quantum signal processing. Last but not least, we explore the very recent extension of these ideas to the SU(2n)-setting. The lecture assumes familiarity with basic (linear) Fourier analysis but aims to be accessible to a broad audience.