Quantum Simulation of Partial Differential Equations via Schrodingerisation - Lecture 1

Explore quantum simulation techniques for solving partial differential equations through the innovative Schrodingerisation method in this comprehensive lecture. Learn how to transform linear PDEs into higher-dimensional Schrödinger equations, enabling their simulation on quantum devices that naturally obey Schrödinger dynamics. Discover the fundamental principles behind quantum simulation, starting with Feynman's original vision from the 1980s and progressing to modern applications that could overcome the curse of dimensionality plaguing classical computational methods. Master the versatile Schrodingerisation approach that works with both discrete-variable quantum systems (qubits) and continuous quantum degrees of freedom (qumodes), offering a more natural representation for PDEs without requiring initial discretization. Understand how this methodology extends beyond basic linear PDEs to encompass systems of linear ordinary differential equations, linear PDEs with random coefficients for uncertainty quantification, and iterative linear algebra problems transformed into evolutionary ODEs. Examine the quantum analog computing aspects that make this approach particularly suitable for near-term quantum device implementation, and explore potential applications to certain nonlinear PDEs. Gain insights into the mathematical foundations that enable D-dimensional linear PDEs to be mapped onto (D+1)-qumode quantum systems, opening new research directions in quantum computational mathematics.