Quantum simulation of partial differential equations: Applications and detailed analysis

We study a recently introduced simple method [S. Jin, N. Liu, and Y. Yu, Quantum simulation of partial differential equations via Schr\"odingerisation, arXiv:2212.13969] for solving general linear partial differential equations with quantum simulation. This method converts linear partial differential equations into a Hamiltonian system, using a simple transformation called the warped phase transformation. Here we provide a more-in-depth technical discussion and expand on this approach in a more detailed and pedagogical way. We apply this to examples of partial differential equations, including heat, convection, Fokker-Planck, linear Boltzmann, and Black-Scholes equations. This approach can also be extended to general linear partial differential equations, including the Vlasov-Fokker-Planck equation and the Liouville representation equation for nonlinear ordinary differential equations. Extension to higher-order time derivatives is also possible.