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.
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.
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.
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.
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
The paper presents a variational quantum algorithm to solve initial–boundary value problems described by second-order partial differential equations. The approach uses hybrid classical/quantum framework that is well suited for quantum computers of the current noisy intermediate-scale quantum era. The partial differential equation is initially translated into an optimal control problem with a modular control-to-state operator (ansatz). The objective function and its derivatives required by the optimizer can efficiently be evaluated on a quantum computer by measuring an ancilla qubit, while the optimization procedure employs classical hardware. The focal aspect of the study is the treatment of boundary conditions, which is tailored to the properties of the quantum hardware using a correction technique. For this purpose, the boundary conditions and the discretized terms of the partial differential equation are decomposed into a sequence of unitary operations and subsequently compiled into quantum gates. The accuracy and gate complexity of the approach are assessed for second-order partial differential equations by classically emulating the quantum hardware. The examples include steady and unsteady diffusive transport equations for a scalar property in combination with various Dirichlet, Neumann, or Robin conditions. The results of this flexible approach display a robust behavior and a strong predictive accuracy in combination with a remarkable polylog complexity scaling in the number of qubits of the involved quantum circuits. Remaining challenges refer to adaptive ansatz strategies that speed up the optimization procedure.
Quantum computation is a promising approach to solving problems in physics, optimization, and numerical simulation. Quantum algorithms leverage superposition and entanglement in an exponentially large Hilbert space to encode and solve problems that are intractable for classical computers. However, these algorithms require error-corrected quantum hardware, which is not yet available. This thesis explores two alternative quantum-based approaches that promise to achieve better scalability than traditional algorithms in the near-term: Hybrid variational quantum algorithms (VQAs) and fully classical, quantum-inspired algorithms. In the first part, we review and benchmark optimization methods well suited for VQAs, specifically the Simultaneous Perturbation Stochastic Approximation (SPSA). We introduce complex-variable extensions and improvements to the second-order and quantum natural implementations. Finally, we assess their performance across quantum control, quantum tomography, and the variational quantum eigensolver, providing practical guidelines for common VQA scenarios. In the second part, we apply Matrix Product States (MPS) to solve time-dependent partial differential equations (PDEs). By integrating Hermite Distributed Approximating Functionals (HDAF) into the MPS framework, we develop a novel encoding to efficiently approximate differential operators with exponential precision. We benchmark this approach against traditional schemes for simulating the expansion of a levitated nanoparticle. The quantum-inspired numerical integration schemes deliver large memory savings and adequate runtimes even in presence of moderate chirping.
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.