Efficient quantum simulation of partial differential equations - Nana Liu

What kinds of scientific computing problems are suited to be solved on a quantum device with quantum advantage? It turns out that by transforming a partial differential equation (PDE) into a higher-dimensional space, certain important issues can be resolved while at the same time not incurring a curse of dimensionality, when tackled by a quantum algorithm.

I'll introduce a simple new way - called Schrodingerisation - that transforms any linear partial differential equation into a set of Schrodinger's equations. This allows one to simulate any general linear partial differential equation via quantum simulation.

Furthermore, Schrodingerisation can also be applied to quantum dynamics in the presence of artificial boundary condition. This method can also be applied to problems in linear algebra by transforming iterative methods in linear algebra into evolution of ordinary differential equations, like the Jacobi method and the power method. This allows us to directly use quantum simulation to solve the linear system of equations and find maximum eigenvectors and eigenvalues of a given matrix.

I'll also explore ways in which quantum algorithms can be used to efficiently solve not just linear PDEs but also certain classes of nonlinear PDEs, like nonlinear Hamilton-Jacobi equations and scalar hyperbolic equations, which are useful in many areas like optimal control and machine learning. PDEs with uncertainties can also be tackled more efficiently with quantum algorithms.