This paper studies the performance of the variational quantum linear solver (VQLS) with an improved ansatz for discretized reservoir flow equations for the first time. First, we introduce the two typical flow equations in reservoir simulation, namely, the diffusion equation for pressure and the convection-dominated Buckley–Leverett equation for water saturation, and their commonly used finite volume or finite difference-based discretized linear equations. Then, we propose an improved ansatz in VQLS to enhance the convergence and accuracy of VQLS and a strategy of adjusting grid order to reduce the complexity of the quantum circuit for preparing the quantum state corresponding to the coefficient vector of the discretized reservoir flow equations. Finally, we apply the modified VQLS to solve the discretized reservoir flow equations by employing the Xanadu's PennyLane open-source library. Four numerical examples are implemented, and the results show that VQLS can calculate reservoir flow equations with high accuracy, and the improved ansatz significantly outperforms the original one. Moreover, we study the effects of reservoir heterogeneity, the number of ansatz layers, the equation type, and the number of shots on the computational performance. Limited by the current computing capacity, the number of grids subject to the involved number of quantum bits in the implemented examples is small; we will further explore this quantum algorithm to practical examples that require a large number of quantum bits in the future.
This paper studies the performance of the variational quantum linear solver (VQLS) with an improved ansatz for discretized reservoir flow equations for the first time. First, we introduce the two typical flow equations in reservoir simulation, namely, the diffusion equation for pressure and the convection-dominated Buckley–Leverett equation for water saturation, and their commonly used finite volume or finite difference-based discretized linear equations. Then, we propose an improved ansatz in VQLS to enhance the convergence and accuracy of VQLS and a strategy of adjusting grid order to reduce the complexity of the quantum circuit for preparing the quantum state corresponding to the coefficient vector of the discretized reservoir flow equations. Finally, we apply the modified VQLS to solve the discretized reservoir flow equations by employing the Xanadu's PennyLane open-source library. Four numerical examples are implemented, and the results show that VQLS can calculate reservoir flow equations with high accuracy, and the improved ansatz significantly outperforms the original one. Moreover, we study the effects of reservoir heterogeneity, the number of ansatz layers, the equation type, and the number of shots on the computational performance. Limited by the current computing capacity, the number of grids subject to the involved number of quantum bits in the implemented examples is small; we will further explore this quantum algorithm to practical examples that require a large number of quantum bits in the future.
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.
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.
Fluid dynamics workloads are dominated by repeated solves of large, structured linear systems, motivating the search for quantum acceleration. The Variational Quantum Linear Solver (VQLS) is a leading near-term candidate, but practical deployment on hybrid quantum--high--performance computing (HPC) systems faces three persistent challenges: (i) the linear-combination-of-unitaries (LCU) encoding of the system matrix explodes in memory and runtime as the problem size grows, (ii) ansatz selection is largely empirical, with no clear link between standard circuit metrics and solver convergence, and (iii) end-to-end VQLS pipelines have rarely been exercised on production HPC hardware at non-trivial qubit counts. This work addresses these challenges through three contributions. First, we benchmark four matrix-encoding strategies---naive LCU, PennyLane-integrated, Fast Walsh--Hadamard Transform (FWHT)-based parallel Pauli decomposition, and an singular value decomposition (SVD)-based two-term LCU---and show that the FWHT approach reduces peak memory by up to 1298× on an 11×11 Hele--Shaw grid, while the SVD-based coherent VQLS delivers over 10,000× per-iteration speedup over standard Pauli-based VQLS at 8 qubits. Second, we evaluate 11 ansatz families with gradient-free and gradient-based optimizers on canonical Hele--Shaw flow, and find that expressibility and entanglement metrics correlate only weakly with VQLS convergence, motivating problem-aware ansatz design. Third, we deploy the full workflow on the OLCF Frontier supercomputer and successfully simulate a 15-qubit tridiagonal Toeplitz system on a single node. Together, these results establish a practical baseline for VQLS in hybrid quantum--HPC computation fluid dynamic (CFD) workflows and identify the remaining bottlenecks for larger problems.
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.
The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm leads a promising tool for solving partial differential equations on noisy intermediate scale quantum devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still at the early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of the VQLS implementation are discussed by various test instances of linear systems. The effect of the number of shots on the accuracy is studied, which reveals a logarithmic relationship. Furthermore, the heuristic scaling of the VQLS with the precision ε, the number of qubits n and the condition number k validates its time complexity reported in the literature. In addition, the successful state vector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present VQLS-based algorithm by proof-of-concept results. Finally, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision ε and linearly dependent on the number of qubits n.
Time-fractional diffusion equations have emerged as powerful models for describing anomalous transport phenomena in physics, biology and engineering. To address the computational challenges arising from their non-local operators, we employ the WEB-spline finite element method, which provides a flexible and accurate discretization framework. The resulting linear system of equations are then explored in the context of quantum computing. Specifically, we investigate three prominent quantum linear solvers: the variational quantum linear solver (VQLS), the Harrow-Hassidim-Lloyd (HHL) algorithm and quantum annealing (QA). VQLS leverages hybrid variational techniques and shallow circuits, making it well-suited for noisy intermediate-scale quantum (NISQ) devices, while HHL offers a theoretically exponential speedup for sparse systems but requires deep fault-tolerant circuits. QA, in contrast, reformulates the problem into a quadratic unconstrained binary optimization (QUBO) instance, enabling approximate solutions through energy minimization on specialized hardware. We present a comparative analysis in terms of circuit depth, noise resilience, scalability and solution extraction, including the role of quantum state tomography in reconstructing classical information. Numerical experiments on a time-fractional diffusion problem highlight the complementary strengths and limitations of each method. This study bridges advanced numerical discretization with emerging quantum algorithms, providing insights into the feasibility and future potential of quantum-enhanced solvers for fractional partial differential equations.