Incompressible Navier-Stokes solve on noisy quantum hardware via a
  hybrid quantum-classical scheme
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-fidelity results can be achieved via this approach, even on
current quantum devices. Multigrid preconditioning of the Poisson problem helps
avoid local minima and reduces resource requirements for the quantum device. A
quantum state readout technique called HTree is used for the first time on a
physical problem. Htree is appropriate for real-valued problems and achieves
linear complexity in the qubit count, making the Navier-Stokes solve further
tractable on current quantum devices. We compare the quantum resources required
for near-term and fault-tolerant solvers to determine quantum hardware
requirements for fluid simulations with complexity improvements.
Zhixin Song Robert Deaton Bryan Gard Spencer H. Bryngelson