Variational Quantum Algorithms for Solving the Diffusion and Burgers' Equation

Abstract

We present variational quantum algorithms to solve general (linear and nonlinear) partial differential equations (PDEs) based on the Feynman-Kitaev Hamiltonian formalism. The method was originally developed for the variational time evolution of quantum systems by Barison et al. [Phys. Rev. Research 4, 043161 (2022)]. It makes use of a clock quantum register to get the full time evolution with a single optimization routine. We generalize the approach to solve classical PDEs and benchmark its performance on applications including diffusion and the creation of shockwaves in Burgers' equation. Our analysis is carried out using a statevector simulator as well as the IBM Q System One quantum computer in Ehningen in Germany. Additionally we corroborate the validity of the variational framework for solving nonlinear PDEs on spacetime grids of various sizes via imaginary time evolution.