Speaker
Description
We present a quantum framework for solving a large class of elliptic and parabolic Partial Differential Equations (PDEs) endowed with periodic conditions. We solve the Poisson equation $\Delta u = f$ and the heat equation on the $d$-dimensional flat torus using a Fourier spectral method implemented on quantum circuits.
The main contribution is an efficient use of block encoding to load the inverse diagonal spectral filter into a quantum circuit. Unlike standard quantum algorithms based on general matrix inversion, our approach constructs the reciprocal of the diagonal entries directly. This procedure requires only one ancillary qubit and can be implemented in polylogarithmic time with respect to the system size $N$.
Different strategies are adopted for solving the problem, depending on the type of equation. For elliptic equations, the solution is obtained by directly applying the inverse spectral filter to the source term. For parabolic equations, we employ an iterative implicit scheme where the spatial variables are handled via our quantum spectral method, while the update for the next time step is performed classically. We validate our approach by
comparing the energy evolution and steady-state solutions against standard classical spectral schemes. The numerical results demonstrate that our quantum solver achieves high fidelity, with errors comparable to machine precision limits.