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In this work, we tackle the resolution of partial differential equations (PDEs) on digital quantum computers. Two fundamental PDEs are addressed: the anisotropic diffusion equation and the anisotropic convection equation. We present a quantum numerical scheme consisting of three steps: quantum state preparation, evolution with diagonal operators, and measurement of observables of interest. The evolution step relies on a high-order centered finite difference and a product formula approximation, also known as Trotterization. We provide novel vector-norm analysis to bound the different sources of error. We prove that the number of time-steps required in the evolution can be reduced by a factor \(\Theta (16^n)\) for the diffusion equation, and \(\Theta (4^n)\) for the convection equation, where n is the number of qubits per dimension, an exponential reduction compared to the previously established operator-norm analysis.

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Quantum Algorithm for Anisotropic Diffusion and Convection Equations with Vector Norm Scaling

  • 会议论文

  • 发表日期：

    2026年

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  • Julien Zylberman,
  • Thibault Fredon,
  • Nuno F. Loureiro,
  • Fabrice Debbasch

摘要

In this work, we tackle the resolution of partial differential equations (PDEs) on digital quantum computers. Two fundamental PDEs are addressed: the anisotropic diffusion equation and the anisotropic convection equation. We present a quantum numerical scheme consisting of three steps: quantum state preparation, evolution with diagonal operators, and measurement of observables of interest. The evolution step relies on a high-order centered finite difference and a product formula approximation, also known as Trotterization. We provide novel vector-norm analysis to bound the different sources of error. We prove that the number of time-steps required in the evolution can be reduced by a factor \(\Theta (16^n)\) for the diffusion equation, and \(\Theta (4^n)\) for the convection equation, where n is the number of qubits per dimension, an exponential reduction compared to the previously established operator-norm analysis.