Sequential water flood simulation alternates implicit pressure solves with explicit saturation updates. This study examines, under ideal statevector access, how a solution state produced by the Harrow‑Hassidim‑Lloyd quantum linear system algorithm can be inserted into that sequential workflow for a two‑phase model in one dimension. The evolving pressure systems are preconditioned from the left using rebuilt sparse approximate inverses, converted by Hermitian block dilation, and treated with signed phase estimation at finite resolution. The formulation also identifies the classical normalization, sign recovery, and complete field readout operations required before a solution state can drive the transport update. Steady and transient component tests precede coupled calculations on three successively refined grids. Across these grids, the statevector and direct pressure solutions remain closely aligned in direction, while the finest grid saturation trajectories show the largest difference and the statevector branch reaches the imposed upper saturation bound. Each plotted statevector pressure is aligned by a global scalar with the exact solution of its current dilated system, so the results assess algebraic data exchange rather than an independent reconstruction of physical pressure. A separate pilot covering five steps finds similar directional agreement between pressure solutions when a right sparse approximate inverse is rebuilt or updated using a sparse approximate map, but the combined setup problem. The results show that close directional agreement between pressure solutions does not by itself guarantee accurate saturation transport and thus provide a reference point for assessing such solvers in sequential reservoir simulation. All calculations use ideal statevectors; no quantum hardware, speedup of the complete workflow, or hardware advantage is demonstrated. Document Type: Original article Cited as:Xu, L., Xiang, Y., Li, M., Zhang, L., Jia, C. Coupled simulation of pressure and saturation in water flooding using a quantum linear system solver. Advances in Geo‑Energy Research, 2026, 21(3): 220‑237. https://doi.org/10.46690/ager.2026.09.03
Sequential water flood simulation alternates implicit pressure solves with explicit saturation updates. This study examines, under ideal statevector access, how a solution state produced by the Harrow‑Hassidim‑Lloyd quantum linear system algorithm can be inserted into that sequential workflow for a two‑phase model in one dimension. The evolving pressure systems are preconditioned from the left using rebuilt sparse approximate inverses, converted by Hermitian block dilation, and treated with signed phase estimation at finite resolution. The formulation also identifies the classical normalization, sign recovery, and complete field readout operations required before a solution state can drive the transport update. Steady and transient component tests precede coupled calculations on three successively refined grids. Across these grids, the statevector and direct pressure solutions remain closely aligned in direction, while the finest grid saturation trajectories show the largest difference and the statevector branch reaches the imposed upper saturation bound. Each plotted statevector pressure is aligned by a global scalar with the exact solution of its current dilated system, so the results assess algebraic data exchange rather than an independent reconstruction of physical pressure. A separate pilot covering five steps finds similar directional agreement between pressure solutions when a right sparse approximate inverse is rebuilt or updated using a sparse approximate map, but the combined setup problem. The results show that close directional agreement between pressure solutions does not by itself guarantee accurate saturation transport and thus provide a reference point for assessing such solvers in sequential reservoir simulation. All calculations use ideal statevectors; no quantum hardware, speedup of the complete workflow, or hardware advantage is demonstrated. Document Type: Original article Cited as:Xu, L., Xiang, Y., Li, M., Zhang, L., Jia, C. Coupled simulation of pressure and saturation in water flooding using a quantum linear system solver. Advances in Geo‑Energy Research, 2026, 21(3): 220‑237. https://doi.org/10.46690/ager.2026.09.03
Abstract Though generic quantum computers are not yet available, we have access to the Noisy Intermediate-Scale Quantum (NISQ) era. The advent of variational quantum algorithms has opened doors for quantum computing in science and engineering during the NISQ era. This study integrates the quantum algorithm with classical streamline methods for efficient, high-precision simulation of two-phase flows, aiming to inform future quantum computing-based reservoir simulation technologies. We employ the variational quantum algorithm to solve the linearized finite volume discrete pressure equations. This process involves decomposing the coefficient matrix of the linear equations using the Pauli basis and preparing the quantum state of the coefficient vector through a unitary operation. A parameterized hardware-efficient ansatz is then constructed, and the quantum circuit’s output (i.e., the cost function value) is obtained via the Hadamard Test. Classical optimizer minimizes the cost function, updating the ansatz parameters to solve the pressure equations. Streamline distribution across the computational domain and time of flight distribution along each streamline are rapidly derived, with water saturation distribution calculated using a high order weighted essentially non-oscillatory (WENO) scheme on each streamline. This results in an integrated workflow combining quantum computing, streamline tracking, and high-order numerical methods. We tested three cases, including homogeneous reservoirs, heterogeneous reservoirs, and multi-well injection and production scenarios. Xanadu’s Pennylane open-source library was used to implement the variational quantum algorithm for computing pressure distribution. The results show that this variational quantum algorithm can achieve high-precision calculation of pressure distribution within fewer optimization steps, with relative computational errors all within 1%. Based on the streamline distribution obtained from tracking, the WENO scheme on the streamlines reduces the numerical dispersion error of the calculated saturation distribution compared to the upwind finite difference scheme, thereby further improving the computational resolution of the waterflooding front. This work pioneers a hybrid quantum-classical workflow for streamline-based reservoir simulation, showcasing its potential for accuracy, efficiency, and robustness in two-phase flow simulations across various reservoir types, paving the way for future quantum computing-based general-purpose reservoir simulators.
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.
Abstract The application of quantum algorithms to classical problems is generally accompanied by significant bottlenecks when transferring data between quantum and classical states, often negating any intrinsic quantum advantage. Here we address this challenge for a well-known algorithm for linear systems of equations, originally proposed by Harrow, Hassidim and Lloyd (HHL), by adapting it into a predictor–corrector instead of a direct solver. Rather than seeking the solution at the next time step, the goal now becomes determining the change between time steps. This strategy enables the intelligent omission of computationally costly steps commonly found in many classical algorithms, while simultaneously mitigating the notorious readout problems associated with extracting solutions from a quantum state. Random or regularly performed skips instead lead to simulation failure. We demonstrate that our methodology secures a useful polynomial advantage over a conventional application of the HHL algorithm. The practicality and versatility of the approach are illustrated through applications in various fields such as smoothed particle hydrodynamics, plasma simulations, and reactive flow configurations. Moreover, the proposed algorithm is well suited to run asynchronously on future heterogeneous hardware infrastructures and can effectively leverage the synergistic strengths of classical as well as quantum compute resources.
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.
The development of quantum processors for practical fluid flow problems is a promising yet distant goal. Recent advances in quantum linear solvers have highlighted their potential for classical fluid dynamics. In this study, we evaluate the Harrow–Hassidim–Lloyd (HHL) quantum linear systems algorithm (QLSA) for solving the idealized Hele–Shaw flow. Our focus is on the accuracy and computational cost of the HHL solver, which we find to be sensitive to the condition number, scaling exponentially with problem size. This emphasizes the need for preconditioning to enhance the practical use of QLSAs in fluid flow applications. Moreover, we perform shots-based simulations on quantum simulators and test the HHL solver on superconducting quantum devices, where noise, large circuit depths, and gate errors limit performance. Error suppression and mitigation techniques improve accuracy, suggesting that such fluid flow problems can benchmark noise mitigation efforts. Our findings provide a foundation for future, more complex application of QLSAs in fluid flow simulations.
Accurate modelling of fluid flow in porous media is crucial for applications including hydrology, geological CO₂ storage, hydrogen storage, geothermal energy, and hydrocarbon recovery. The inherent heterogeneity of subsurface formations, characterized by spatial permeability contrasts, fractures, and vugs, leads to large and often ill-conditioned linear systems when discretized at high resolution. Although upscaling techniques reduce computational cost, they may obscure fine-scale features that significantly influence flow behavior, particularly in the presence of small but highly conductive fractures. This study investigates the Variational Quantum Linear Solver (VQLS) as an alternative framework for solving discretized Darcy flow equations in one-, two-, and three-dimensional (1D, 2D, and 3D) homogeneous and heterogeneous systems. VQLS approximates the solution to linear systems of the form Ax = b by preparing a quantum state proportional to the classical solution vector. The framework incorporates a hybrid quantum-classical cost formulation, which strengthens optimization in ill-conditioned systems, and a curvature-aware Adaptive Dynamic Ansatz, which enables principled circuit depth selection without manual tuning. Five experimental modes are evaluated, including fixed-depth training, hybrid cost formulation, adaptive ansatz growth, depth sweep analysis, and hybrid weight sweep. All experiments are conducted using noiseless statevector simulation. The results examine convergence behavior, solution accuracy, and algorithmic limitations in heterogeneous media, providing insight into the practical potential and current constraints of variational quantum solvers for porous media flow modelling.