An SU(2n)-valued nonlinear Fourier transform

TL;DR

This paper introduces a nonlinear Fourier transform mapping matrix sequences to $SU(2n)$-valued functions, characterizes its image, constructs an inverse, and connects it to quantum signal processing.

Contribution

It defines a new nonlinear Fourier transform for matrix sequences, characterizes its image, and links it to quantum signal processing over $U(2n)$.

Findings

Characterization of the transform's image for finitely supported and square-summable sequences.

Construction of an inverse transform for certain $SU(2n)$-valued functions.

Application to quantum signal processing over $U(2n)$.

Abstract

We define a nonlinear Fourier transform which maps sequences of contractive $n \times n$ matrices to $SU(2n)$-valued functions on the circle $\mathbb{T}$. We characterize the image of finitely supported sequences and square-summable sequences on the half-line, and construct an inverse for $SU(2n)$-valued functions whose diagonal $n \times n$ blocks are outer matrix functions. As an application, we relate this nonlinear Fourier transform with quantum signal processing over $U(2n)$ and multivariate quantum signal processing.