Incompressible flow simulation via a hybrid quantum-classical approach and variational algorithm

Partial differential equation (PDE) solvers for Navier-Stokes fluids problems are important for many science and engineering problems. Quantum algorithms have been verified to solve linear PDEs and verified on simulators. Fault-tolerant quantum hardware may make exponential speedups possible, for example, via linearization methods and HHL-type linear solvers. Still, NISQ-era near-term quantum hardware requires algorithms that demand less quantum volume: shallower gate depths and fewer qubits. Variational algorithms, like VQE and VQLS, are appropriate under such restrictions. Here, we present work on a hybrid approach to solving the nonlinear incompressible Navier-Stokes equations. A classical computer performs the nonlinear computations, and a quantum algorithm, on simulator or hardware, performs the cumbersome Poisson equation solve that enforces mass continuity. A lid-driven cavity problem is investigated at various Reynolds numbers and grid resolutions to determine the sensitivity of the global and local Poisson equation to the variation algorithm and quantum noise.