论文标题
用可微分量子电路求解非线性微分方程
Solving nonlinear differential equations with differentiable quantum circuits
论文作者
论文摘要
我们提出了一种量子算法来求解非线性微分方程的系统。使用量子特征图编码，我们将函数定义为参数化量子电路的期望值。我们使用自动分化来表示分析形式的功能衍生物作为可区分的量子电路（DQC），从而避免了计算梯度的有限差异程序。我们描述了一个混合量子古典工作流程，其中训练DQC以满足微分方程和指定的边界条件。作为一个特定的示例设置，我们展示了这种方法如何在高维特征空间中实现求解微分方程的光谱方法。从技术角度来看，我们设计了Chebyshev量子特征图，该图提供了强大的拟合多项式的基础集，并具有丰富的表达性。我们模拟算法以求解Navier-Stokes方程的实例，并在收敛性喷嘴中计算流体流量的密度，温度和速度曲线。
We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations, and compute density, temperature and velocity profiles for the fluid flow in a convergent-divergent nozzle.
