Solving non-linear PDEs on a quantum computer

本项目涉及在量子计算机上求解非线性偏微分方程。最近，研究表明，在经典计算机上求解非线性偏微分方程的传统多网格（MG）方法可以通过MG重整化（MGR）方法获得指数级加速[Journal of Computational Physics， 372, 587(2018)]。在这个项目中，博士生将在量子计算机上实现MGR方法，专注于流体动力学中的工业相关问题。根据量子硬件未来可用性的当前时间表（例如，通过我们与IBM的合作），我们预计我们的量子算法将在未来3 - 5年内超越经典计算机。由于非线性偏微分方程在所有科学和技术领域都无处不在，因此我们的方法有可能在广泛的现实世界应用中利用量子增强算法。主要任务是：(i)在现有量子器件上开发求解非线性偏微分方程的算法的优化实现，旨在实现指数级加速。这需要开发特定于平台的算法，以减轻它们各自的弱点，从而获得量子优势。该学生将考虑基于捕获离子的量子设备，特别是Q20:20机器，以及IBM量子计算网络上的设备。（ii）确定量子计算机上的MGR方法可以帮助解决突出问题或提高现有解决方案质量的工业相关应用。该项目的这一部分将与BAE系统公司密切合作，BAE系统公司对解决由非线性偏微分方程建模的复杂流体动力学问题感兴趣。

This project is concerned with solving non-linear partial differential equations (PDEs) on a quantum computer. Recently, it has been shown that an exponential speedup in conventional multigrid (MG) methods for solving non-linear PDEs on a classical computer can be obtained via MG renormalization (MGR) methods [Journal of Computational Physics 372, 587 (2018)]. In this project, the DPhil student will implement MGR methods on a quantum computer focussing on industrially relevant problems in fluid dynamics. Based on current timetables for the future availability of quantum hardware (e.g. through our partnership with IBM), we expect our quantum algorithms to outperform classical computers within the next 3 - 5 years. Since non-linear PDEs are ubiquitous in all areas of science and technology, our method has the potential to leverage the use of quantum-enhanced algorithms in a broad range of real-world applications. The main tasks are:(i) developing optimized implementations of algorithms for solving non-linear PDEs on existing quantum devices aiming to achieve an exponential speedup. This entails developing platform-specific algorithms mitigating their individual weaknesses such that a quantum advantage can be obtained. The student will consider quantum devices based on trapped ions and in particular the Q20:20 machine, as well as devices on IBM's quantum computing network.(ii) identifying industrially relevant applications where MGR methods on a quantum computer could help solving outstanding problems or improve the quality of existing solutions. This part of the project will be carried out in close collaboration with BAE Systems, who are interested in solving complex fluid dynamics problems that are modelled by non-linear PDEs.