Abstract: Efficient quantum algorithms for linear ODEs have now been developed. Some of them achieve optimal query complexity with respect to the coefficient matrices, yet it remains an open problem how to leverage these algorithms to solve linear PDEs.

We give detailed analysis and circuit design of structure-preserving quantum algorithms for second-order linear evolutionary PDEs, including parabolic equations and hyperbolic equations with mixed Dirichlet, Neumann, and periodic boundary conditions and source terms. Our method-of-lines approach investigates the boundary lifting and boundary-aware discretization, so that the resulting semi-discrete systems are stable and compatible with efficient quantum ODE primitives. For the parabolic problem, we use a diagonal similarity transform to ensure the semi-discrete generator have a positive semi-definite Hermitian part, and then solve the resulting ODEs by the optimal LCHS. For the hyperbolic problem, we rewrite the semi-discrete equation as an equivalent first-order system and solve it by Hamiltonian simulation. Finally we implement our quantum algorithms with explicit block-encoding constructions and circuit implementations, as well as demonstrating the end-to-end complexity bounds.

Biography: Yixuan Liang is a Ph.D. student at Qiuzhen College, Tsinghua University, supervised by Prof. Jin-Peng Liu (YMSC).