Trahan Corey Jason, Loveland Mark, Davis Noah, Ellison Elizabeth
Information and Technology Laboratory, U.S. Army Engineer Research and Development Center, Vicksburg, MS 39180, USA.
Applied Research Laboratories, The University of Texas at Austin, Austin, TX 78713, USA.
Entropy (Basel). 2023 Mar 28;25(4):580. doi: 10.3390/e25040580.
Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.
有限元方法是求解偏微分方程数值解的行业标准。然而，即使对于现代超级计算机，应用规模仍然是这些方法实际应用的关键。例如，大型多尺度应用可能会受到其对极其庞大的线性系统解的需求的限制。因此，研究近期的量子算法是否有可能比经典线性求解器具有任何优势是值得的。在本研究中，我们研究了最近提出的用于求解偏微分方程离散解的变分量子线性求解器（VQLS）。发现该方法与线性系统规模呈多对数缩放，并且该方法可以在有噪声的中等规模量子（NISQ）计算机上使用浅量子电路来实现。在此，我们利用混合VQLS来求解稳态泊松方程以及与时间相关的热方程和波动方程。
