标题： 哈密顿量模拟用于具有任意输运场的对流扩散方程 显示英文标题

标题： Hamiltonian Simulation for Advection-Diffusion Equation with arbitrary transport field

摘要： 我们提出了一种新颖的方法，通过量子启发的“薛定谔化”技术来求解在任意传输场下的对流-扩散方程。 尽管存在许多求解偏微分方程（PDE）的方法，但哈密顿量模拟仍然是一个相对未被充分探索但前景广阔的领域——尤其是在长期容错量子计算的背景下。 基于这一潜力，我们的量子算法旨在适应非平凡的空间变化传输场，并适用于二维和三维的对流-扩散问题。 为了确保数值稳定性和准确性，该算法结合了对流部分的迎风离散化方案和扩散部分的中心差分法，通过定制的近似和优化技术组合，使其适用于量子实现。 我们在涉及二维和三维耦合旋转、剪切和扩散传输的基准场景中展示了该算法的有效性。 此外，我们在IBM量子硬件上使用16个量子比特实现了二维对流-扩散方程，验证了我们的方法，并突显了其实际适用性和鲁棒性。 显示英文摘要

摘要： We present a novel approach to solve the advection-diffusion equation under arbitrary transporting fields using a quantum-inspired 'Schrodingerisation' technique for Hamiltonian simulation. Although numerous methods exist for solving partial differential equations (PDEs), Hamiltonian simulation remains a relatively underexplored yet promising direction-particularly in the context of long-term, fault-tolerant quantum computing. Building on this potential, our quantum algorithm is designed to accommodate non-trivial, spatially varying transport fields and is applicable to both 2D and 3D advection-diffusion problems. To ensure numerical stability and accuracy, the algorithm combines an upwinding discretization scheme for the advective component and the central differencing for diffusion, adapted for quantum implementation through a tailored mix of approximation and optimization techniques. We demonstrate the algorithm's effectiveness on benchmark scenarios involving coupled rotational, shear, and diffusive transport in two and three dimensions. Additionally, we implement the 2D advection-diffusion equation using 16 qubits on IBM Quantum hardware, validating our method and highlighting its practical applicability and robustness.