Leong Fong Yew, Ewe Wei-Bin, Koh Dax Enshan
Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), Singapore, 138632, Singapore.
Sci Rep. 2022 Jun 25;12(1):10817. doi: 10.1038/s41598-022-14906-3.
Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. Here, we propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. The use of encoded source states informed by preceding solution vectors results in faster convergence compared to random re-initialization. Through statevector simulations of the heat equation, we demonstrate how the time complexity of our algorithm scales with the Ansatz volume for gradient estimation and how the time-to-solution scales with the diffusion parameter. Our proposed algorithm extends economically to higher-order time-stepping schemes, such as the Crank-Nicolson method. We present a semi-implicit scheme for solving systems of evolution equations with non-linear terms, such as the reaction-diffusion and the incompressible Navier-Stokes equations, and demonstrate its validity by proof-of-concept results.
变分量子算法为在近期量子计算机上求解偏微分方程提供了一种很有前景的新范式。在此，我们提出一种变分量子算法，通过拉普拉斯算子的隐式时间步长法来求解一般的演化方程。与随机重新初始化相比，使用由先前解向量提供信息的编码源状态会导致更快的收敛。通过热方程的态矢量模拟，我们展示了我们算法的时间复杂度如何随用于梯度估计的量子近似优化算法（Ansatz）体积而缩放，以及求解时间如何随扩散参数而缩放。我们提出的算法可以经济地扩展到高阶时间步长方案，如克兰克 - 尼科尔森方法。我们提出了一种半隐式方案来求解具有非线性项的演化方程组，如反应扩散方程和不可压缩纳维 - 斯托克斯方程，并通过概念验证结果证明了其有效性。
