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Explore quantum simulation techniques for solving partial differential equations through the innovative Schrodingerisation method in this comprehensive mathematical lecture. Learn how to transform linear PDEs into higher-dimensional Schrödinger equations, enabling their simulation on quantum devices that naturally follow quantum mechanical principles. Discover the theoretical foundations behind mapping classical PDEs onto quantum systems, moving beyond the traditional focus on Schrödinger's equation to encompass a broader range of mathematical problems. Examine the versatility of this approach across both discrete-variable quantum systems (qubits) and continuous quantum degrees of freedom (qumodes), with particular emphasis on how the continuous representation offers advantages for PDE simulation without requiring initial discretization. Understand applications to linear PDEs, systems of linear ordinary differential equations, and linear PDEs with random coefficients relevant to uncertainty quantification. Investigate extensions to linear algebra problems through the transformation of iterative methods into linear ODE evolution, and explore potential applications to certain nonlinear PDEs. Engage with current research directions and open questions in quantum simulation methodology, presented as part of the CEMRACS Summer School on Quantum Computing at the Centre International de Rencontres Mathématiques.
