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This research demonstrates a significant methodological advance for NISQ-era quantum computing by transforming hardware noise from an accuracy limit into a mere convergence rate factor. By employing iterative refinement on the pressure-Poisson equation, the study proves that meaningful precision can be achieved without waiting for fault-tolerant devices or achieving near-perfect state fidelity. This approach effectively bridges the gap between current noisy hardware limitations and the practical requirements of solving complex partial differential equations.
Quantum-Classical Iterative Refinement for the Pressure-Poisson Equation on NISQ Hardware
arXiv:2610.24342v1 Announce Type: new Abstract: 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 $\Delta t/\rho$. On the IBM processor ibm_cleveland, a four-qubit ($16\times16$) 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 ($64\times64$) pilot, all three runs reached the same tolerance in 13 corrections. On the IonQ Forte processor, a manufactured-solution benchmark reached $5.6\times10^{-8}$ in nine corrections, with a pressure error of $4.9\times10^{-8}$. 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 $5.6\times10^{-4}$ 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.
Authors:Doyeol "David" Ahn (1,2), Byeongyong Park (3), Rho Shin Myong (4), Franz Klein (5) ((1) Department of Electrical and Computer Engineering, University of Seoul, Seoul, Republic of Korea 2 Singularity Quantum Inc., Newport Beach, CA, USA 3 Center for Quantum Information Processing, University of Seoul, Seoul, Republic of Korea 4 School of Mechanical and Aerospace Engineering, Gyeongsang National University, Jinju, Republic of Korea 5 National Quantum Laboratory at Maryland (QLab), University of Maryland, College Park, MD, USA)
Abstract: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. The hardware noise then sets the rate of convergence but not the attainable accuracy, and it is this change of the role of the quantum solve, not an improvement of the hardware, that allows the residual to converge. With a boundary-consistent staggered (MAC) discretization, the pressure residual equals the velocity divergence left after projection up to the factor $\Delta t/\rho$. On the IBM processor ibm_cleveland, a four-qubit ($16\times16$) 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 ($64\times64$) pilot, all three runs reached the same tolerance in 13 corrections. On the IonQ Forte processor, a manufactured-solution benchmark reached $5.6\times10^{-8}$ in nine corrections. With the variational training itself carried out on ibm_cleveland, the residual fell to $5.6\times10^{-4}$ in seven corrections, below the pre-specified tolerance of $10^{-3}$, at the rate predicted by a noise model. To the best of our knowledge, these are the first converged pressure solutions obtained on quantum hardware.
arXiv:2610.24342 [quant-ph]
(or arXiv:2610.24342v2 [quant-ph] for this version)
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