  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 upcoming quantum hardware presents noise in the computations,
requiring algorithms that make modest use of quantum resources: shallower
circuit depths and fewer qubits. Variational algorithms are more appropriate
and robust under resource restrictions. This work presents a hybrid
quantum-classical algorithm for the incompressible Navier-Stokes equations.
Classical devices perform nonlinear computations, and quantum ones use
variational algorithms to solve 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-fidelity results can be achieved via this approach, even on
current quantum devices. A multigrid preconditioning approach helps avoid local
minima. HTree, a tomography technique with linear complexity in qubit count,
reduces the quantum state readout time. We compare the quantum resources
required for near-term and fault-tolerant solvers to determine quantum hardware
requirements for fluid simulations with complexity improvements.
