Numerical solutions for fluid flow problems are challenging and have been focus of Computational Fluid Dynamics (CFD) research for past several decades. The advent of quantum computing promises exponential speedup in comparison to existing classical methods and alleviate computational constraints posed by CFD problems. Although solutions for most problems of interest in fluid dynamics using quantum computing are distant, recent advances in algorithms, software and hardware provide a path towards realizing this goal. Quantum linear solver algorithms (QLSA) such as Harrow–Hassidim–Lloyd (HHL) and Variational Quantum Linear Solver (VQLS) have been successfully implemented to solve for canonical problems such as Hele-Shaw flow. However, these algorithms still suffer to scale and address problems with ill-conditioned Jacobians. In the current paper, we alleviate these restrictions with a new quantum solver based on Singular Value Decomposition (SVD) and simulate flow past a 2D cylinder. The fidelity of the SVD based quantum solver in predicting the flow past 2D cylinder is computed along with an assessment of errors. Classical and quantum solutions for the flow are compared for different resolutions. Finally, we discuss variation in the solutions based on number of shots used.
Numerical solutions for fluid flow problems are challenging and have been focus of Computational Fluid Dynamics (CFD) research for past several decades. The advent of quantum computing promises exponential speedup in comparison to existing classical methods and alleviate computational constraints posed by CFD problems. Although solutions for most problems of interest in fluid dynamics using quantum computing are distant, recent advances in algorithms, software and hardware provide a path towards realizing this goal. Quantum linear solver algorithms (QLSA) such as Harrow–Hassidim–Lloyd (HHL) and Variational Quantum Linear Solver (VQLS) have been successfully implemented to solve for canonical problems such as Hele-Shaw flow. However, these algorithms still suffer to scale and address problems with ill-conditioned Jacobians. In the current paper, we alleviate these restrictions with a new quantum solver based on Singular Value Decomposition (SVD) and simulate flow past a 2D cylinder. The fidelity of the SVD based quantum solver in predicting the flow past 2D cylinder is computed along with an assessment of errors. Classical and quantum solutions for the flow are compared for different resolutions. Finally, we discuss variation in the solutions based on number of shots used.
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.
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.
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.
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.
Many computational fluid dynamics (CFD) algorithms solve partial differential equations by discretization, resulting in large and sparse systems of linear equations. While iterative methods are widely used to solve these systems classically, most existing quantum linear system solvers target the solution through matrix inversion rather than approximating it using an iterative procedure. In this work, we develop a quantum implementation of the Jacobi method based on quantum singular value transformation (QSVT). By reformulating the Jacobi iteration as a polynomial transformation of a block-encoded operator, the algorithm requires only a constant ancilla overhead with respect to the number of iterations while maintaining a circuit depth that scales linearly with the iteration number. We demonstrate the algorithm for one- and two-dimensional Poisson problems, including the pressure Poisson equation arising in Chorin's projection method for the lid-driven cavity flow. The proposed algorithm provides a promising building block for future quantum implementations of multigrid methods and preconditioning techniques, bringing quantum algorithms closer to established CFD solution strategies.
There has been significant progress in the development of quantum algorithms for solving linear systems of equations with a growing body of applications to Computational Fluid Dynamics (CFD) and CFD-like problems. This work extends previous work by developing a non-linear hybrid quantum-classical CFD solver and using it to generate fully converged solutions. The hybrid solver uses the SIMPLE CFD algorithm, which is common in many industrial CFD codes, and applies it to the 2-dimensional lid driven cavity test case. A theme of this work is the classical processing time needed to prepare the quantum circuit with a focus on the decomposition of the CFD matrix into a linear combination of unitaries (LCU). CFD meshes with up to 65x65 nodes are considered with the largest producing a LCU containing 32,767 Pauli strings. A new method for rapidly re-computing the coefficients in a LCU is proposed, although this reduces, rather than eliminates, the classical scaling issues. The quantum linear equation solver uses the Harrow, Hassidim, Lloyd (HHL) algorithm via a state-vector emulator. Test matrices are sampled from the classical CFD solver to investigate the solution accuracy that can be achieved with HHL. For the smallest 5x5 and 9x9 CFD meshes, full non-linear hybrid CFD calculations are performed. The impacts of approximating the LCU and the varying the number of ancilla rotations in the eigenvalue inversion circuit are studied. Preliminary timing results indicate that the classical computer preparation time needed for a hybrid solver is just as important to the achievement of quantum advantage in CFD as the time on the quantum computer. The reported HHL solutions and LCU decompositions provide a benchmark for future research. The CFD test matrices used in this study are available upon request.