An SU(2n)-valued nonlinear Fourier transform

TL;DR

This work extends the nonlinear Fourier transform to $SU(2n)$-valued functions by mapping contractive $n\times n$ matrices to $SU(2n)$-valued functions on the circle and characterizing the image for finitely supported and square-summable right-half-line sequences. Central contributions include a matrix inner/outer theory with Szegő spectral factorization, a constructive RH-factorization-based inverse for outer diagonal blocks, and a comprehensive $\ell^2$ theory with layer stripping that establishes a homeomorphism between data on the half-line and the NLFT image. The framework yields a robust Plancherel identity and Lipschitz stability results, and connects to quantum signal processing (QSP) and multivariate QSP through concrete factorization and mapping properties. Together, these results provide a rigorous, scalable higher-dimensional NLFT with algorithmic implications for QSP and related quantum algorithms.

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.

Theorems & Definitions (92)

• Definition 1.1
• Theorem 1.2
• Definition 1.3: Forward finite matrix NLFT
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1: wiener
• proof
• Lemma 2.2: Barclay_cty_specESS_Quant_cty_spec
• proof : Proof of Lemma \ref{['lem:cty_spec_factors']}