The Poisson equation has applications across many areas of physics and engineering, such as the dynamic process simulation of ocean current. Here we present a quantum algorithm for solving Poisson equation, as well as a complete and modular circuit design. The algorithm takes the HHL algorithm as the framework (where HHL is for solving linear equations). A more efficient way of implementing the controlled rotation, one of the crucial steps in HHL, is developed based on the arc cotangent function. The key point is that the inverse trigonometric function can be evaluated in a very simple recursive way by a binary expansion method. Quantum algorithms for solving square root and reciprocal functions are proposed based on the classical non-restoring method. These advances not only reduce the algorithm’s complexity, but more importantly make the circuit more complete and practical. We demonstrate our circuits on a quantum virtual computing system installed on the Sunway TaihuLight supercomputer. This is an important step toward practical applications of the present circuits as a fast Poisson solver in the near-term hybrid classical/quantum devices.

中文翻译：

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量子快速泊松求解器：算法以及完整的模块化电路设计

泊松方程在物理和工程学的许多领域都有应用，例如洋流的动态过程仿真。在这里，我们提出一种求解泊松方程的量子算法，以及完整的模块化电路设计。该算法以HHL算法为框架（其中HHL用于求解线性方程）。基于反余切函数，开发了一种更有效的控制旋转方法，这是HHL中的关键步骤之一。关键是，可以通过二进制展开方法以非常简单的递归方式对逆三角函数求值。基于经典的非复原方法，提出了求解平方根和倒数函数的量子算法。这些进步不仅降低了算法的复杂度，但更重要的是使电路更完整，更实用。我们在安装在Sunway TaihuLight超级计算机上的量子虚拟计算系统上演示了我们的电路。这是在本电路的实际应用中迈出的重要一步，作为近期混合经典/量子器件中的快速泊松求解器。