Variational Spacetime Simulation of the 1D Burgers Equation

Solving partial differential equations (PDEs) in fluid dynamics has long been a challenging problem due to nonlinearity, multiscale structure, and the accumulation of stepwise errors in time-marching schemes. In this work, we present a variational quantum approach to the one-dimensional Burgers equation, a canonical PDE that serves as a standard testbed for nonlinear advection and shock-formation, turbulence-like regimes. Our approach encodes the velocity field on a discretized spacetime grid as a quantum state, and the solution is obtained through global optimization. This spacetime formulation can incorporate different kinds of boundary conditions, which enables the compressed sensing scheme. We have evaluated the performance of this approach using classical simulation of quantum circuits across different viscosity settings.