How to represent a function in a quantum computer

Quantum Signal Processing (QSP) is an algorithmic process by which one represents a signal $f: [0,1] \to (-1,1)$ as the upper left entry of a product of $SU(2)$ matrices parametrized by the input variable $x \in [0,1]$ and some ”phase factors” $\{\psi_k\}_{k \geq 0}$ depending on $f$. We show that, after a change of variables, QSP is actually the SU(2)-valued nonlinear Fourier transform, and the phase factors $\{\psi_k\}_k$ correspond to the nonlinear Fourier coefficients. By exploiting a nonlinear Plancherel identity and using some basic spectral theory, we then show that a QSP representation exists for every $f$ satisfying the log integrability condition \[ \int\limits_{0} ^1 \log (1-f(x)^2) \frac{dx}{\sqrt{1-x^2}} > – \infty \, . \]