Quantum fast Poisson solver: the algorithm and complete and modular circuit design

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.