Abstract
Partial differential equation solvers are required to solve the Navier-Stokes equations for fluid flow. Recently, algorithms have been proposed to simulate fluid dynamics on quantum computers. Fault-tolerant quantum devices might enable exponential speedups over algorithms on classical computers. However, current and foreseeable quantum hardware introduce noise into computations, requiring algorithms that make judicious use of quantum resources: shallower circuit depths and fewer qubits. Under these restrictions, variational algorithms are more appropriate and robust. This work presents a hybrid quantum-classical algorithm for the incompressible Navier--Stokes equations. A classical device performs nonlinear computations, and a quantum one uses a variational solver for the pressure Poisson equation. A lid-driven cavity problem benchmarks the method. We verify the algorithm via noise-free simulation and test it on noisy IBM superconducting quantum hardware. Results show that high-fideli