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.

翻译：流体流动的Navier-Stokes方程求解需要借助偏微分方程求解器。近期，已有研究提出在量子计算机上模拟流体动力学的算法。容错量子设备可能实现相对于经典计算机算法的指数级加速。然而，当前及近期的量子硬件存在计算噪声，这要求算法需节制使用量子资源：即采用更浅的电路深度和更少的量子比特。变分算法在资源受限条件下更为适用且鲁棒。本研究提出了一种针对不可压缩Navier-Stokes方程的混合量子-经典算法。经典设备执行非线性计算，而量子设备则采用变分算法求解压力泊松方程。通过顶盖驱动方腔问题对该方法进行基准测试。我们通过无噪声仿真验证算法，并在含噪声的IBM超导量子硬件上进行测试。结果表明，即使基于当前量子设备，该方法仍可获得高保真度结果。采用多重网格预处理方法有助于避免局部极小值。HTree——一种具有量子比特数线性复杂度的层析技术——显著减少了量子态读取时间。通过比较近期与容错求解器所需的量子资源，我们确定了实现复杂度改进的流体模拟所需的量子硬件条件。