Quantum-Classical Iterative Refinement for the Pressure-Poisson Equation on NISQ Hardware

We present a hybrid quantum-classical iterative refinement for the pressure-Poisson equation of the incompressible Navier-Stokes projection method on noisy quantum hardware. Rather than solving the pressure equation once per time step, we solve it repeatedly for the normalized residual and scale each quantum direction by a least-squares fit to the physical residual. As a result, the error of the quantum solve enters as a contraction factor $\sin {\theta }_k$ of the residual rather than as an error floor. This removes the requirement of a state infidelity below ${10}^{-7}$ that was estimated for single-solve schemes. With a boundary-consistent staggered (MAC) discretization, the pressure residual equals the velocity divergence left after projection up to the factor $\mathrm{\Delta }t\mathrm{/}\rho$. On the IBM processor ibm_cleveland, a four-qubit ($16\times 16$) problem reached a relative residual below ${10}^{-7}$ within eight to ten corrections in all 20 runs of a pre-registered experiment, whereas a single solve with a comparable number of shots stopped at 0.33-0.59. In a six-qubit ($64\times 64$) pilot, all three runs reached the same tolerance in 13 corrections. On the IonQ Forte processor, a manufactured-solution benchmark reached $5.6\times {10}^{-8}$ in nine corrections, with a pressure error of $4.9\times {10}^{-8}$. Finally, we carried out the variational training itself on ibm_cleveland, so that every sampled value of the cost function was obtained from hardware measurements. The residual fell to $5.6\times {10}^{-4}$ in seven corrections, below the pre-specified tolerance of ${10}^{-3}$, at the rate predicted by a noise model and with 24 min of QPU time.