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arXiv:2610.24342

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

Doyeol "David" Ahn, Byeongyong Park, Rho Shin Myong, Franz Klein

quant-phSubmitted Oct 17, 2026• Announced Oct 20, 2026• 0 citations• Revised Oct 20, 2026 (v2)
v1 · Oct 17, 2026v2 · Oct 20, 2026
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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 $\sinθ_k$ 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 $Δt/ρ$. 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.

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In computational fluid dynamics (CFD), the numerical integration of the Navier-Stokes equations is frequently constrained by the Poisson equation to determine the pressure. Discretization of this equation often results in the need to solve a system of linear algebraic equations. This step typically represents the primary computational bottleneck. Quantum linear system algorithms such as Harrow-Hassidim-Lloyd (HHL) offer the potential for exponential speedups for solving sparse linear systems, such as those that arise from the discretized Poisson equation. In this work, we successfully couple HHL to a discretized formulation of the incompressible Navier-Stokes equations and demonstrate both accurate lid-driven cavity flow simulations as a fully integrated benchmark problem, and accurate flow of the Taylor-Green vortex. To address the readout limitation, we utilize a recent novel quantum state tomography (QST) approach based on Chebyshev polynomials and Quantum Amplitude Estimation (QAE), which enables approximate statevector extraction without full state reconstruction. Together, these results clarify the algorithmic structure required for quantum CFD, explicitly confront the measurement bottleneck, and establish benchmark problems for future quantum fluid simulations. We implement the solver using IBM's Qiskit framework and validate the hybrid quantum-classical simulation against standard classical numerical methods. Our results demonstrate that the hybrid solver successfully captures the global vortex dynamics of the lid-driven cavity problem and the Taylor-Green vortex, offering a robust pathway for integrating quantum subroutines into more practical higher-Reynolds number CFD workflows.
Quantum linear-system algorithms (QLSAs) have rigorous worst-case complexity guarantees, but their runtimes are often chosen from spectral information assumed in advance. What is largely lacking is an a posteriori progress flag: most QLSA workflows, unlike the classical counterparts, do not provide a built-in mechanism to signal whether a particular instance has already converged. For discretizations of elliptic PDEs −∇⋅(a(x)∇u(x))=f(x),-\nabla\cdot(a(x)\nabla u(x))=f(x),−∇⋅(a(x)∇u(x))=f(x), with divergence--gradient structure −∇⋅(a(x)∇)≈Ah=Gh†Gh, -\nabla\cdot \big(a(x)\nabla) \approx A_h=G_h^\dagger G_h, −∇⋅(a(x)∇)≈Ah​=Gh†​Gh​, we formulate a stable first-order ODE whose limiting solution block is the desired Galerkin solution. The PDE-dependent scale is then \(\norm{G_h}=\bigO(h^{-1})\), comparable to factorized QLSA constructions with square-root condition-number scaling. We design an augmented dynamics with residual variables, in which measuring a residual register gives an on-the-fly convergence indicator without reconstructing the solution vector. For smooth right-hand sides, dynamic stopping can reduce the evolution time and gate count relative to a fixed worst-case schedule, and may also reduce exposure to accumulated hardware errors. Numerical experiments for a two-dimensional finite element Poisson problem show that the residual-register probability follows the actual error decay and, for some right-hand sides, can stop the quantum circuit well before a conservative worst-case runtime estimate is reached.
Near-term quantum algorithms are a promising route to solving partial differential equations, but gauging their true potential requires separating algorithmic performance from sampling and hardware noise. We benchmark a ground-state variational quantum eigensolver (VQE), cast as a variational quantum linear solver, against the Trotterization, variational quantum imaginary time evolution, and adaptive variational quantum dynamics simulation methods applied to the one-dimensional advection-diffusion equation in the recent quantum-dynamics study by Alipanah et al. [Phys. Rev. Res. 7, 043318 (2025)2643-156410.1103/ndc3-bdwt] at matched grid and problem size. On a noiseless state-vector simulator the N=4 VQE drives the final-time infidelity to a numerical floor (∼10^{-14}) once the depth reaches L≈5, an algorithmic ceiling set by exact expectation values. Evaluating the same solver with a finite number S of measurement shots, still without hardware noise, makes the infidelity sampling limited, following 1-f≈c/S (a best-case readout-sampling estimate, with the solution's signs assumed known), providing a regime-matched comparison with the shot-based emulator of Alipanah et al. and explaining the gap to their noisy hardware runs (>10^{-1}). The benchmark thus decomposes the near-term error budget into algorithmic, sampling, and hardware contributions, with a matched-depth resource comparison. The formulation applies without modification across N=4,5,6 qubits and to a two-dimensional (eight-qubit, 16×16) problem evolved to t=1, where the state-vector VQE holds a ∼10^{-7} algorithmic-ceiling infidelity against the sampling-limited ∼10^{-5} of the corresponding shot-based simulation, a difference of measurement regime rather than algorithmic superiority.
Summary In the current noisy intermediate-scale quantum (NISQ) era, the variational quantum algorithm (VQA) opens the door to introducing quantum computing into geosciences. Therefore, this paper will study the application effect of VQA in solving the pressure equations in reservoir flow problems for the first time and provide a reference for constructing standardized field-scale quantum algorithms for reservoir numerical simulation in the future. We first obtain the linearized discrete pressure equations using the standard finite volume procedure, and then use the variational quantum linear solver (VQLS) to solve the discrete equations to obtain the pressure distribution in the reservoir domain. Besides, we propose two practical strategies to reduce the complexity of preparing the quantum state corresponding to the coefficient vector of the pressure equations. Xanadu’s Pennylane open-source library is adopted to implement the hybrid quantum computing procedure of this study. A toy case is given to illustrate the technical detail and the computational performance of the quantum computing procedure. The results show that the workflow can obtain high-accuracy reservoir pressure distribution and the strategies can accelerate the optimization convergence speed to achieve more accurate solutions in the same number of optimization steps.
Variational quantum circuits (VQCs) solving partial differential equations (PDEs) on near-term quantum hardware face a critical challenge: hardware noise degrades solution fidelity and disrupts convergence. We present a systematic study of three noise channels; depolarizing, amplitude damping, and bit-flip on VQCs constrained by PDE residual loss functions for the heat equation, Burgers' equation, and the Saint-Venant shallow water equations. We benchmark three error mitigation strategies: zero-noise extrapolation (ZNE) via Richardson polynomial fitting, probabilistic error cancellation (PEC), and measurement error mitigation through inverse confusion matrices. Our numerical experiments on 6-qubit, 4-layer circuits demonstrate that ZNE reduces absolute error by 82-96% at low noise (p = 0.001), with effectiveness degrading gracefully at higher noise strengths. We prove analytically and confirm numerically that physics-constrained circuits exhibit inherent noise resilience: at p = 0.01, constrained circuits maintain 25-47% higher fidelity than unconstrained counterparts, with the advantage scaling with PDE complexity. PEC provides near-exact correction at low gate counts but incurs exponential sampling overhead, rendering it impractical beyond ~60 gates at p >= 0.02. Error budget decomposition reveals that systematic errors dominate at all noise levels (43-58%), while the PDE residual component grows from ~10% to ~31% as noise increases, indicating that physics constraints absorb noise through structured gradient information. These results establish practical guidelines for deploying variational PDE solvers on NISQ hardware.
Quantum computing has the potential to revolutionize computational problem-solving by leveraging the quantum mechanical phenomena such as superposition and entanglement, enabling fundamentally different computational strategies compared to classical computing. Classical methods for solving complex and high dimensional partial differential equations (PDEs), crucial for modeling diverse physical phenomena, suffer from an exponential growth in computational complexity known as the curse of dimensionality. This can limit their effectiveness for certain high-resolution simulations and real-time computational applications. In contrast, quantum computing inherently exploits superposition and entanglement to efficiently represent and simultaneously manipulate large-scale discretized PDEs, using exponentially fewer resources compared to classical systems. There are currently many quantum techniques available for solving partial differential equations (PDEs), which are mainly based on variational quantum circuits. However, the existing quantum PDE solvers, particularly those based on variational quantum eigensolver (VQE) techniques, suffer from several limitations. These include low accuracy, high execution times, and low scalability on quantum simulators as well as on noisy intermediate-scale quantum (NISQ) devices, especially for multidimensional PDEs. In this work, we propose an efficient and scalable algorithm for solving multidimensional PDEs. We present two variants of our algorithm: the first leverages finite-difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, while the second employs FDM, C2Q, and column-by-column decomposition (CCD). Both variants are designed to enhance accuracy and scalability while reducing execution times. We have validated and evaluated our proposed concepts using a number of case studies including multidimensional Poisson equation, multidimensional heat equation, Black Scholes equation for financial option pricing, and Navier-Stokes equation for computational fluid dynamics (CFD), achieving promising results. Our results demonstrate higher accuracy, higher scalability, and faster execution times compared to VQE-based solvers on noise-free and noisy quantum simulators from IBM. Additionally, we validated our approach on hardware emulators and actual quantum hardware, employing noise mitigation techniques. This work establishes a practical and effective approach for solving PDEs using quantum computing for engineering and scientific applications.