Solving non-linear PDEs on a quantum computer

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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.