泊松方程量子求解算法

Quantum Algorithms for Solving Poisson Equations

地球上海洋和大气的运动规律遵从Navier-Stokes(NS)方程的约束,该方程及其导出的地球流体动力学方程组属于非线性偏微分方程组,通常采用数值方法求解,而随着离散网格数目的增加,求解该方程已逐渐超出目前经典超算的计算能力。量子计算是一种基于量子力学的基本原理进行信息处理的新一代计算模式,相比于经典计算具有指数加速能力,使用量子计算机有望破解全球海洋和大气动力学模拟的算力困境。本文对该课题作了初期探索,即在一定约束条件下,NS方程的求解可规约到泊松方程求解上;针对不同情境,本文设计了三种泊松方程量子求解算法,它们相对经典计算均具有加速效果。本文给出了三种求解算法的完全模块化的量子运行线路、能在现有量子计算机上直接运行演示。本文提出的泊松方程量子求解算法可以直接应用于其他科学研究领域。

The movement rules of oceans and atmosphere on the earth are dictated by the Navier-Stokes(NS)equation.This equation and its derived earth-fluid dynamic system equations belong to nonlinear partial differential equations,which are usually solved by the numerical method.With the increase of numbers of discrete grid cells,to solve these equations needs capacity far beyond the computing power of current supercomputers.Quantum computing is a new generation computing mode for information processing based on the basic principles of quantum mechanics,and it can provide exponential speed-up capability comparing to the classical counterparts.Thus,the application of quantum computer may provide us a way to break the bottleneck of computing power for simulating global ocean and atmospheric dynamics.In this paper,a preliminary exploration is made.Since the solution of NS equation can be reduced to the solution of Poisson equations under certain constraint conditions,here we have developed three kinds of quantum algorithms,for solving Poisson equations,which have speed-up effect compared with the classical methods.We have provided the complete and modular quantum circuits for the three quantum algorithms which can be run directly by the present quantum computer.The present quantum algorithms for solving Poisson equations can be applied directly in other research fields.