AI 中文总结
该工作提出一种基于量子奇异值变换的量子Jacobi迭代算法，以恒定辅助比特开销和线性电路深度求解线性方程组，并成功应用于一维及二维Poisson方程，为量子多网格方法奠定基础。
AI 中文摘要
许多计算流体力学（CFD）算法通过离散化求解偏微分方程，从而产生大型稀疏线性方程组。虽然迭代方法在经典计算中被广泛用于求解这些方程组，但大多数现有的量子线性系统求解器旨在通过矩阵求逆来求解，而非使用迭代过程进行近似。在本工作中，我们基于量子奇异值变换（QSVT）开发了Jacobi方法的量子实现。通过将Jacobi迭代重新表述为块编码算子的多项式变换，该算法在迭代次数方面仅需恒定的辅助比特开销，同时保持电路深度随迭代次数线性扩展。我们针对一维和二维Poisson问题演示了该算法，包括Chorin投影法中用于顶盖驱动空腔流动的压力Poisson方程。所提出的算法为未来多网格方法和预处理技术的量子实现提供了有前景的构建模块，使量子算法更接近成熟的CFD求解策略。
英文摘要
Many computational fluid dynamics (CFD) algorithms solve partial differential equations by discretization, resulting in large and sparse systems of linear equations. While iterative methods are widely used to solve these systems classically, most existing quantum linear system solvers target the solution through matrix inversion rather than approximating it using an iterative procedure. In this work, we develop a quantum implementation of the Jacobi method based on quantum singular value transformation (QSVT). By reformulating the Jacobi iteration as a polynomial transformation of a block-encoded operator, the algorithm requires only a constant ancilla overhead with respect to the number of iterations while maintaining a circuit depth that scales linearly with the iteration number. We demonstrate the algorithm for one- and two-dimensional Poisson problems, including the pressure Poisson equation arising in Chorin's projection method for the lid-driven cavity flow. The proposed algorithm provides a promising building block for future quantum implementations of multigrid methods and preconditioning techniques, bringing quantum algorithms closer to established CFD solution strategies.
Comments15 pages, 6 figures
