CCF: AF: Small: Simulating Hamiltonian dynamics: Algorithms and applications

模拟量子系统的动力学是量子计算机的一个主要潜在应用。 例如，量子模拟可以用于模拟化学反应和预测材料的行为。 它也可以用作开发其他量子算法的工具，它的研究揭示了量子计算机的力量。 本项目主要研究量子模拟的两个相关方面：为基本模拟问题开发改进的量子算法和研究量子模拟在其他计算任务中的应用。虽然量子模拟算法的技术水平相当先进，但这一基本任务的最终限制仍不清楚。 该项目在稀疏矩阵描述的量子系统的广泛适用框架内运行，旨在在所有相关模拟参数之间进行最佳权衡。 它还研究了用于模拟开放量子系统的量子算法，利用来自无噪声系统的深入研究的算法思想，为与环境相互作用的噪声量子系统的更现实的设置提供新的算法。 此外，该项目还研究了量子模拟算法的应用。 量子模拟是量子算法的核心组成部分，用于提取有关大型线性方程组解的信息。 该项目将最近的改进应用于量子模拟算法，以改进线性系统的量子算法。 它还研究了特定类型的线性系统的计算能力，例如在计算电网络中的有效电阻时出现的那些。 最后，该项目研究了如何将高级量子模拟算法的算法思想应用于模拟环境之外，例如，在计算数论中。

Simulating the dynamics of quantum systems is a major potential application of quantum computers. Quantum simulation can be applied, for example, to model chemical reactions and to predict the behavior of materials. It can also be used as a tool to develop other quantum algorithms, and its study sheds light on the power of quantum computers. This project focuses on two related aspects of quantum simulation: developing improved quantum algorithms for basic simulation problems and studying applications of quantum simulation to other computational tasks.While the state of the art in quantum simulation algorithms is fairly advanced, the ultimate limits for this fundamental task remain unclear. Operating within a broadly applicable framework of quantum systems described by sparse matrices, the project aims to develop optimal tradeoffs among all relevant simulation parameters. It also studies quantum algorithms for simulating open quantum systems, leveraging algorithmic ideas from the well-studied scenario of noiseless systems to give new algorithms for the more realistic setting of noisy quantum systems that interact with their environments. Furthermore, the project studies applications of quantum simulation algorithms. Quantum simulation is the core component of a quantum algorithm for extracting information about the solutions of large systems of linear equations. The project applies recent improvements to quantum simulation algorithms to give improved quantum algorithms for linear systems. It also studies the computational power of particular types of linear systems, such as those that arise in the computation of effective resistances in electrical networks. Finally, the project investigates how algorithmic ideas from advanced quantum simulation algorithms can be applied outside the context of simulation, e.g., in computational number theory.