The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it restricts the mesh size since the solution of the linear system becomes computationally intractable when the mesh resolution increases. The reliance on the ability of High-Performance Computers (HPC) to scale up and meet these requirements is myopic; such very high-fidelity simulations require a paradigm shift in computing. This paper presents an exploration of quantum methodologies aimed at achieving high accuracy in solving such a large system of equations. Leveraging recent works in Quantum Linear Solver Algorithms (QLSA) and variational algorithms suitable for Quantum Simulation in HPC, we aspire to push the boundaries of CFD-relevant problems that can be solved on hybrid quantum-classical framework. To this end, we consider the 2D, transient, incompressible, viscous, non-linear coupled Burgers equation as a test problem and investigate the accuracy of our approach by comparing results with a classical linear system of equation solvers, such as the Generalized Minimal RESidual method (GMRES). Through rigorous testing, our findings demonstrate that our quantum methods yield results comparable in accuracy to traditional approaches. Additionally, we demonstrate the accuracy, scalability, and consistency of our quantum method. Lastly, we present an insightful estimation of the resources our quantum algorithm needs to solve systems with nearly 2 billion mesh points.
The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it restricts the mesh size since the solution of the linear system becomes computationally intractable when the mesh resolution increases. The reliance on the ability of High-Performance Computers (HPC) to scale up and meet these requirements is myopic; such very high-fidelity simulations require a paradigm shift in computing. This paper presents an exploration of quantum methodologies aimed at achieving high accuracy in solving such a large system of equations. Leveraging recent works in Quantum Linear Solver Algorithms (QLSA) and variational algorithms suitable for Quantum Simulation in HPC, we aspire to push the boundaries of CFD-relevant problems that can be solved on hybrid quantum-classical framework. To this end, we consider the 2D, transient, incompressible, viscous, non-linear coupled Burgers equation as a test problem and investigate the accuracy of our approach by comparing results with a classical linear system of equation solvers, such as the Generalized Minimal RESidual method (GMRES). Through rigorous testing, our findings demonstrate that our quantum methods yield results comparable in accuracy to traditional approaches. Additionally, we demonstrate the accuracy, scalability, and consistency of our quantum method. Lastly, we present an insightful estimation of the resources our quantum algorithm needs to solve systems with nearly 2 billion mesh points.
As the energy sector transitions towards more sustainable forms of energy production, classical Computational Fluid Dynamics (CFD) increasingly struggles with the bottle- necks of simulating high complexity turbulent flows. While quantum computing (QC) theoretically offers an exponential speedup for large-scale linear systems, current Noisy Intermediate-Scale Quantum (NISQ) hardware lacks the error correction required for deep and complex circuits that mathematically ideal algorithms require. Furthermore, the inherent linearity of quantum mechanics seriously impedes the solving non-linear partial differential equations (PDEs), which are critical for fluid modeling. To bridge thisgap, thisthesisevaluatestheVariationalQuantumLinearSolver(VQLS),a shallow- circuit, hybrid algorithm, applied to the 1D viscous Burgers’ equation while utilizing the Cole-Hopf transformation to analytically linearize the equation. In both simulated and actual hardware, the VQLS algorithm is implemented across increasingly scaled topographies (2, 4, and 8 qubits). This work intends to identify the limitations of implementing quantum fluid dynamics in near-term quantum 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.
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.
Quantum computing (QC) represents an emerging frontier with significant potential for transformative advancements in Computational Fluid Dynamics (CFD). This potential has been explicitly recognized by the CFD Vision 2030 report, emphasizing the strategic importance of exploring QC for future breakthroughs. Among quantum algorithms relevant to CFD, the Variational Quantum Linear Solver (VQLS) is particularly promising due to its suitability for solving linear and Hamiltonian-based problems, characteristic of fundamental CFD equations such as the pressure-Poisson equation. However, the practical implementation of VQLS for large-scale CFD problems faces critical scalability challenges, most notably the exponential increase in the number of quantum circuit evaluations and the barren plateau phenomenon, which severely hampers trainability and algorithm convergence. In this study, two novel strategies are introduced to address these challenges. First, we utilize a block encoding technique based on efficient oracles, thereby significantly reducing quantum circuit complexity and qubit overhead while accessing system matrix. Second, we incorporate a quantum-inspired evolutionary optimization (QIEO) approach, demonstrating enhanced training efficiency and effectively mitigating the barren plateau issue. These improvements are integrated within a Hybrid Quantum-Classical Method (HQCM) linear solver framework and demonstrated through solving a 2D steady-state heat equation, representative of the pressure-Poisson equations common in incompressible CFD simulations. The proposed methodologies yield promising results, showcasing improved circuit efficiency, enhanced stability during optimization, and accurate numerical solutions. This work lays a foundational step toward realizing practical, scalable quantum-assisted CFD solvers and sets clear pathways for future research aimed at further algorithmic and architectural enhancements.
To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated computational fluid dynamic algorithms coupled with powerful supercomputers can not find approximate solutions for several design challenges in both adequate time and scale-resolving accuracy. One difficulty arises from solving high dimensional, strongly nonlinear partial differential equations, such as the Navier-Stokes equations, which capture the underlying physics. For nearly all classical algorithms, methods closely related to Newton's method are used to approximate a solution to the problem. Approximately solving the large-scale linear systems of equations occurring in this iterative scheme is generally a main contributor to the total computational complexity. In this paper a new quantum linear system solver supporting Newton's classical method to solve nonlinear partial differential equations is introduced. We present a new variant of the HHL algorithm, requiring less apriori information regarding the eigenvalues of the corresponding matrix. We apply this quantum linear system solver in a hybrid quantum-classical fashion to solve nonlinear partial differential equations. Moreover, a resource estimation for advanced use-cases of practical relevance is provided. Our results demonstrate how quantum computation may improve existing classical methodologies for solving nonlinear partial differential equations. This approach provides another promising application of quantum computers and presents a possible way forward for handling nonlinearities on inherently linear quantum systems.
Recent advancements in quantum computing provide opportunities to address the challenges of computational resource limitations in computational fluid dynamics (CFD). This work presents a hybrid quantum-classical CFD framework that leverages quantum linear algorithms to practical flow simulations. By transforming nonlinear fluid dynamics problems into linear systems, the framework employs quantum linear algorithms to compute solutions, effectively integrating quantum and classical computing approaches. Furthermore, this framework employs subspace methods to map the original large-scale linear systems to small ones solved by quantum linear algorithms, thereby enabling the solution of large-scale problems using the currently limited quantum resources. We applied the famous Harrow–Hassidim–Lloyd (HHL) algorithm and variational quantum linear solver to simulate complex flows, including unsteady flows around a cylinder, simulation of an aircraft with turbulence model, and combustion flows. The performance and quantum resource consumption of these algorithms were evaluated in practical flow scenarios, demonstrating their effectiveness with an average relative error lower than 0.001%. We conducted quantum resource and convergence analyses to adapt this framework for use with near-term quantum computers. Our framework supports simulations of up to 9.4 × 106 grid cells, marking a significant advancement from previous quantum approaches limited to simple flows. This paper not only offers a way for employing quantum linear algorithms in solving complex flow problems but also provides insights into optimizing quantum algorithms for CFD applications, pushing forward the practical utilization of quantum CFD.