15/04/2026 - Julien ZYLBERMAN (CERFACS)

Abstract

The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, two fundamental PDEs are addressed: the anisotropic diffusion equation and the anisotropic convection equation. A quantum numerical scheme consisting of three steps (quantum state preparation, evolution with diagonal operators, and measurement of observables of interest) is presented. The evolution step relies on a high-order centered finite difference and a product formula approximation, also known as Trotterization. A vector-norm analysis is provided to bound the different sources of error. We prove that the number of time-steps can be reduced by a factor exponential in the number of qubits compared to previously established operator-norm analysis, thereby significantly lowering the projected computational resources. We also present efficient quantum circuits and numerical simulations that confirm the predicted vector-norm scaling. Additionally, we report results on real quantum hardware for the one-dimensional convection equation.References:[1]Julien Zylberman et al 2026 Quantum Sci. Technol. 11 025015 ; [2]Zylberman, J., Fredon, T., Loureiro, N.F., Debbasch, F. (2026). Quantum Algorithm for Anisotropic Diffusion and Convection Equations with Vector Norm Scaling, Quantum Engineering Sciences and Technologies for Industry and Services. QUEST-IS 2025. Communications in Computer and Information Science, vol 2744. Springer, Cham. https://doi.org/10.1007/978-3-032-13855-2_23