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arXiv:2601.03987

An SU(2n)-valued nonlinear Fourier transform

Michel Alexis, Lars Becker, Diogo Oliveira e Silva, Christoph Thiele

math.CAmath.FAquant-phSubmitted Jan 7, 2026• Announced Jan 8, 2026• Revised Mar 23, 2026 (v2)
v1 · Jan 7, 2026v2 · Mar 23, 2026
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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.

Comments
Corrected acknowledgements, main article unchanged. Still 54 pages plus references and glossary

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The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on SU(1,1)\mathrm{SU}(1,1)SU(1,1) has been widely studied, its SU(2)\mathrm{SU}(2)SU(2) variant has only recently attracted attention due to emerging applications in quantum signal processing (QSP) and quantum singular value transformation (QSVT). In this paper, we investigate the inverse NLFT on SU(2)\mathrm{SU}(2)SU(2) and establish the numerical stability of the layer stripping algorithm for the first time under suitable conditions. Furthermore, we develop a fast and numerically stable algorithm, called inverse nonlinear fast Fourier transform, for performing inverse NLFT with near-linear complexity. This algorithm is applicable to computing phase factors for both QSP and the generalized QSP (GQSP).
A pivotal algorithm in quantum signal processing represents a real-valued function as a specific entry in a product of SU(2)SU(2)SU(2) matrices, where an argument matrix alternates with a sequence of coefficient matrices. Recently, this framework has been identified as a nonlinear Fourier series, a concept studied across various mathematical disciplines. This Arbeitsgemeinschaft explored the intersection of quantum computing and harmonic analysis, emphasizing the potential of this connection.
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Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on U ( N ) , which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on U ( 2 ) in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate N -interval decision achieving O ( d ) query complexity with a log 2 ⁡ N improvement over iterative U ( 2 ) -QSP requiring O ( d log 2 ⁡ N ) queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements.