AI 中文总结
研究利用量子计算机求解卡尔曼线性化伯格斯方程的挑战，采用非酉线性组合方法加载方程，用变分量子线性求解器结合多重网格方法求解，移植大量离散化点电路到真实硬件，有望产生量子优势。
AI 中文摘要
利用量子计算机有效求解非线性常微分方程和偏微分方程是一项重大挑战，因其固有线性。为应对此挑战，提出卡尔曼线性化方法将非线性常微分方程转化为线性方程组，以便应用现有量子线性系统算法求解。然而，该方法也带来诸多挑战。本文应对了其中一些挑战，能在真实和模拟量子硬件上求解卡尔曼线性化一维伯格斯方程。所有模拟在BlueQubit平台进行，可在GPU或QPU上无缝运行量子电路。首先证明可用非酉线性组合方法将方程有效加载到量子计算机上，再用变分量子线性求解器求解线性系统。因朴素实现受贫瘠高原现象阻碍，引入多重网格方法分阶段求解，前一阶段解作为下一阶段热启动，显著提高了解的精度。最后将时空离散化点总数达\(2^{80} \approx 10^{24}\)的电路移植到真实量子硬件上，表明该方法未来可能产生量子优势。
英文摘要
Efficiently solving nonlinear ordinary and partial differential equations using a quantum computer is a major challenge due its inherent linearity. To circumvent this challenge, the Carleman linearization method has been proposed to transform a nonlinear ordinary differential equation into a linear system of equations, the primary advantage being that existing quantum linear systems algorithms may then be applied to obtain a solution. However, this methodology also brings forth several major challenges that must be addressed to attain a quantum advantage. Herein, we address several of these challenges enabling us to solve the Carleman linearized one-dimensional Burgers' equation on real and simulated quantum hardware. All simulations were performed on BlueQubit's platform allowing for quantum circuits to be run on GPU or QPU's seamlessly. We first demonstrate that the Carleman linearized Burgers' equation can be efficiently loaded onto a quantum computer using the linear combination of non-unitaries method, an alternative to the linear combintaiton of unitaries approach. Once loaded, the linear system is then solved using the variational quantum linear solver. Since a naive implementation of this solver is hindered by the barren plateau phenomenon, we introduce a multigridding method to solve the problem in a series of stages with the solution of the previous stage acting as a warm start for the next stage. This approach is found to significantly improve the accuracy of the solution compared with a naive cold start. Finally, circuits with a combined number of spatial and temporal discretization points totaling up to $2^{80} \approx 10^{24}$ are transpiled onto real quantum hardware demonstrating that the proposed methodology could feasibly produce a quantum advantage on future hardware.
Comments31 pages, 17 figures
