Towards Quantum Speedup for Solving High-Dimensional Partial Differential Equations

高维偏微分方程的高效模拟一直是许多科学领域的核心任务之一。量子技术和算法的最新进展表明，量子算法可以成为一种新的工具来克服维度诅咒，与经典实现相比，量子算法具有实现指数级加速的潜力。这个项目的目标是研究量子算法在高效求解高维微分方程组方面的潜在应用。利用量子力学的力量，新提出的量子算法和技术有望显著加快此类高维偏微分方程组的模拟速度，并给出与空间网格总数呈多对数关系和空间维度呈多项式关系的代价。拟议项目的发展将为克服PDE模拟中的维度诅咒提供新的前景，推动为微分方程设计的最先进的量子算法，并有助于为后量子科学计算铺平道路。在教育方面，学生将在数学和量子信息科学方面得到良好的跨学科培训。该项目旨在为量子和经典问题开发高效的高维微分方程组的量子算法，并建立严格的误差界和复杂性估计，并识别能够和不能有效地量化处理的问题。这样的高维微分方程包括应用于分子动力学的薛定谔方程，以及其他经典的微分方程，如生物应用中出现的反应扩散方程。将解决以下具体方面的问题。对于量子动力学模拟，我们的目标是处理无界算符的动力学模拟，我们探索了诸如向量范数尺度分析、相互作用图中的量子高振荡协议以及解决问题的多尺度方面的半经典/微局部分析等技术。对于可能是非么正的经典动力学，我们提出并探索了使用块编码预言的时间推进策略，旨在提供关于刚性微分方程的量子算法的教学描述，准确地指出量子算法设计与经典数值分析之间的差异。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Efficient simulation of high-dimensional partial differential equations has been one of the core tasks in many scientific areas. Recent advances of quantum technologies and algorithms revealed that quantum algorithms can be a new tool to overcome the curse of dimensionality, with the potential of achieving exponential speedups compared to classical implementations. The goal of this project is to investigate potential applications of quantum algorithms on efficiently solving high-dimensional differential equations. Taking advantage of the power of quantum mechanics, the newly proposed quantum algorithms and techniques are expected to significantly accelerate the simulation of such high-dimensional PDEs and give a cost depending poly-logarithmically on the total number of spatial grids and polynomially on the spatial dimension. The development of proposed projects will provide new prospects to overcome the curse of dimensionality in PDE simulations, to advance the state-of-the-art quantum algorithm designed for differential equations, and to help pave the path towards post-quantum scientific computing. On the educational side, the students involved will get good interdisciplinary training in both mathematics and quantum information science.This project aims to develop efficient quantum algorithms for high-dimensional differential equations for both quantum and classical problems, and to establish rigorous error bounds and complexity estimates, and identify the problems that can and can not be efficiently handled quantumly. Such high-dimensional differential equations include the Schrodinger equation with applications to molecular dynamics, and other classical differential equations, such as reaction-diffusion equations emerging from biological applications. The following specific aspects will be addressed. For quantum dynamics simulation, the goal is to deal with dynamics simulation with unbounded operators, we explore techniques such as the vector norm scaling analysis, quantum highly oscillatory protocol in the interaction picture, and semiclassical/microlocal analysis addressing the multiscale aspects of the problem. For classical dynamics that can be non-unitary, we propose and explore time-marching strategies using block encoding oracles, and aim to provide a pedagogical description on quantum algorithms for stiff differential equations, pinpointing the differences between quantum algorithm design and classical numerical analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.