Towards Quantum Speedup for Solving High-Dimensional Partial Differential Equations

迈向求解高维偏微分方程的量子加速

• 批准号: 2347791
• 负责人: Di Fang
• 金额: $ 15万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-10-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2347791&HistoricalAwards=false
• 关键词: Towards; Quantum; Speedup; Solving; Dimensional

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.

高维偏微分方程的有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，量子算法可以成为克服维数灾难的新工具，与经典实现相比，具有实现指数加速的潜力。本计画的目标是探讨量子演算法在高效率求解高维微分方程式上的潜在应用。利用量子力学的力量，新提出的量子算法和技术有望显着加快这种高维偏微分方程的模拟，并给出一个成本依赖于多项式的空间网格的总数和多项式的空间维度。拟议项目的发展将提供新的前景，以克服PDE模拟中的维数灾难，推进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，参与的学生将获得良好的数学和量子信息科学的跨学科培训，该项目旨在为量子和经典问题的高维微分方程开发高效的量子算法，建立严格的误差界和复杂性估计，并确定可以和不可以有效地量子处理的问题。这种高维微分方程包括薛定谔方程与分子动力学的应用，和其他经典微分方程，如反应扩散方程出现的生物应用。将讨论以下具体方面。对于量子动力学模拟，我们的目标是处理与无界算子的动力学模拟，我们探索的技术，如向量范数标度分析，量子高振荡协议的相互作用图片，和半经典/微局域分析解决多尺度方面的问题。对于可能是非幺正的经典动力学，我们提出并探索了使用块编码预言机的时间推进策略，旨在为刚性微分方程的量子算法提供教学描述，该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估，更广泛的影响审查标准。