High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the spatial dimension, a quantum computer can produce a quantum state that encodes the solution with only polynomial complexity, provided that suitable input access is available. In this work, we investigate efficient quantum algorithms for linear and nonlinear fractional reaction-diffusion equations with periodic boundary conditions. For linear equations, we analyze and compare the complexity of various methods, including the second-order Trotter formula, time-marching method, and truncated Dyson series method. We also present a novel algorithm that combines the linear combination of Hamiltonian simulation technique with the interaction picture formalism, resulting in optimal scaling in the spatial dimension. For nonlinear equations, we employ the Carleman linearization method and propose a block-encoding version that is appropriate for the dense matrices that arise from the spatial discretization of fractional reaction-diffusion equations.

翻译：高维分数阶反应-扩散方程在生物学、化学和物理学领域具有广泛应用，并展现出丰富的现象。经典算法在空间维度上具有指数复杂度，而量子计算机在提供适当输入访问的前提下，能以仅多项式复杂度的量子态编码解。本文研究具有周期边界条件的线性和非线性分数阶反应-扩散方程的高效量子算法。对于线性方程，我们分析并比较了多种方法的复杂度，包括二阶特罗特公式、时间推进方法和截断戴森级数方法。我们还提出了一种新算法，将哈密顿模拟的线性组合技术与相互作用图形式相结合，在空间维度上实现了最优缩放。对于非线性方程，我们采用卡尔曼线性化方法，并提出一种块编码版本，该版本适用于分数阶反应-扩散方程空间离散化产生的稠密矩阵。