Marek Arsenault

Two distinct paradigms for quantum functional programming have been developed in the last few years: Quantum Signal Processing (QSP)-based methods, including the Quantum Singular Value Transformation (QSVT) [GSLW19], and methods based on higher-order transformations, such as the Universal Hamiltonian Eigenvalue Transformation (UHET) [OKTM25]. 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) [MW24]. Building on this result, we introduce a linearised variant of QSVT, which we call Universal Hamiltonian Singular Value Transformation (UHSVT). This algorithm enables a transformation by any sufficiently differentiable function f of any arbitrary matrix A encoded in a block of a Hamiltonian, whose dynamics is accessible as a black box, and it can be performed in a wider range of settings than previous QSVT-based approaches. [GSLW19] A Gilyén, Y Su, G H Low, N Wiebe, STOC, 2019. [OKTM25] T Odake, H Kristjánsson, P Taranto, M Murao, PRR, 2025. [MW24] D Motlagh, N Wiebe, PRX Quantum, 2024.