Linearised quantum signal processing

• URL: http://arxiv.org/abs/2608.14387v1
• Date: Fri, 14 Aug 2026 15:29:37 GMT
• Title: Linearised quantum signal processing
• Abstract summary: We show that UHET can be interpreted as a (randomised) linearisation of Generalised QSP (GQSP)<n>We introduce a linearised variant of (Hamiltonian-based) QSVT, that enables the efficient transformation of the singular values of any arbitrary matrix $A$ encoded in a block of a Hamiltonian.<n>Our algorithm requires the sole condition that $f$ vanishes at the origin.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Quantum functional programming has been developed through two distinct paradigms in the last few years: Quantum Signal Processing (QSP)-based methods, including the Quantum Singular Value Transformation (QSVT), and methods based on higher-order quantum transformations, such as the Universal Hamiltonian Eigenvalue Transformation (UHET). While UHET performs functional transformations of Hamiltonian dynamics, its relationship to QSP-based techniques has remained unclear despite evident structural similarities. In this work, we resolve this gap by establishing a connection between UHET and QSP-based frameworks; specifically, we show that UHET can be interpreted as a (randomised) linearisation of Generalised QSP (GQSP). Building on this result, we introduce a linearised variant of (Hamiltonian-based) QSVT, which we call Universal Hamiltonian Singular Value Transformation (UHSVT), that enables the efficient transformation of the singular values of any arbitrary matrix $A$ encoded in a block of a Hamiltonian, whose dynamics is accessible as a black box, by any sufficiently differentiable complex-valued function $f$. Our algorithm requires the sole condition that $f$ vanishes at the origin, in contrast to previous QSVT-based approaches that assumed either a lower bound on the singular values of $A$ or the ability to perform $X$-rotation gates on the induced two-dimensional 'qubitised' subspace.

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