This paper explores the potential of quantum computing for computational fluid dynamics (CFD) applications, with a focus on turbomachinery CFD. A harmonic balance solver is developed for Burgers’ equation, in which discrete Fourier transforms are approximated using a hybrid quantum-classical algorithm based on the quantum Fourier transform (QFT). Three novel algorithms are presented to estimate Fourier coefficients, providing complete knowledge of their amplitudes and phases, which is not possible with the standard QFT. Their behaviour is studied theoretically and numerically, under noiseless and noisy conditions. The algorithms, whose performance is limited by the extensive sampling required to achieve a small error on the Fourier coefficients, tolerate low levels of depolarising noise. The behaviour of the hybrid solver is investigated for a baseline case with a Reynolds number of 1000, 7 harmonics and 100 grid cells, using the best performing algorithm in a noiseless setting with up to 10 8 samples per QFT. Residuals decrease until errors introduced by statistical uncertainty dominate the total error on the solution. Nevertheless, the hybrid solutions match the classical one closely, with RMS residuals as low as 10 –4 . The impact of several solver parameters on convergence and solution quality is also assessed, including the effect of noise. Although the latter rapidly degrades the solution, the solver achieves satisfactory approximations to the classical solution with 0.001% of depolarising noise. Without seeking to demonstrate a quantum advantage, this work offers valuable insights into the opportunities and challenges of quantum computing, helping readers understand how to design, implement and study quantum algorithms, as well as evaluate their impact in CFD applications.
This paper explores the potential of quantum computing for computational fluid dynamics (CFD) applications, with a focus on turbomachinery CFD. A harmonic balance solver is developed for Burgers’ equation, in which discrete Fourier transforms are approximated using a hybrid quantum-classical algorithm based on the quantum Fourier transform (QFT). Three novel algorithms are presented to estimate Fourier coefficients, providing complete knowledge of their amplitudes and phases, which is not possible with the standard QFT. Their behaviour is studied theoretically and numerically, under noiseless and noisy conditions. The algorithms, whose performance is limited by the extensive sampling required to achieve a small error on the Fourier coefficients, tolerate low levels of depolarising noise. The behaviour of the hybrid solver is investigated for a baseline case with a Reynolds number of 1000, 7 harmonics and 100 grid cells, using the best performing algorithm in a noiseless setting with up to 10 8 samples per QFT. Residuals decrease until errors introduced by statistical uncertainty dominate the total error on the solution. Nevertheless, the hybrid solutions match the classical one closely, with RMS residuals as low as 10 –4 . The impact of several solver parameters on convergence and solution quality is also assessed, including the effect of noise. Although the latter rapidly degrades the solution, the solver achieves satisfactory approximations to the classical solution with 0.001% of depolarising noise. Without seeking to demonstrate a quantum advantage, this work offers valuable insights into the opportunities and challenges of quantum computing, helping readers understand how to design, implement and study quantum algorithms, as well as evaluate their impact in CFD applications.
Abstract Quantum computing (QC) has emerged as a new computational paradigm with the potential for exponential speedup in specific applications. However, current limitations of quantum technology, including decoherence, noise, and restricted qubit counts, pose significant challenges. In this work, we present a proof-of-concept for a hybrid quantum-classical Harmonic Balance (h-HB) solver for time-periodic flow problems. The method leverages the Quantum Fourier Transform (QFT) for Fourier coefficient computation, motivated by its theoretical exponential time-complexity advantage over the Fast Fourier Transform (FFT). We derive error bounds, analyze convergence, and directly compare the hybrid solver’s performance with a classical HB solver for a time-periodic flow problem modeled by the 2D Burgers equation. While practical quantum advantage remains beyond reach, our results demonstrate that the hybrid solver qualitatively reproduces classical solutions. This work provides valuable insights into quantum CFD for periodic flow problems, highlights key challenges, and lays the groundwork for future research toward solving the Navier-Stokes equations with quantum-assisted methods.
Quantum linear solvers are well studied as standalone quantum algorithms; however, in hybrid classical-quantum routines, their practical value must be evaluated at the level of the full non-linear application. A central issue is whether the approximation error of the quantum linear solver remains controlled once embedded in a full iterative workflow. We study this question in the context of a hybrid computational fluid dynamics (CFD) scheme. Through numerical simulations, we analyze how an approximate quantum linear solver affects the convergence of the overall CFD iteration. We show that convergence can be preserved for a non-exact quantum solver with only a moderate overhead in iteration count, provided that the high-frequency components of the linear system are resolved with sufficient accuracy. In addition, we develop an approximate qubitization-based solver (Cheb-LCU) that can reduce quantum resource requirements relative to a Quantum Singular Value Transformation (QSVT)-based solver while inducing only a small loss in convergence performance. This claim is demonstrated through explicit implementation and compilation of the quantum algorithms and by examining their impact on the convergence of the full CFD scheme. We find that our approximate approach reduces the required number of single-qubit rotations by over an order of magnitude relative to the QSVT-based solver, while requiring only a modest increase in CFD iteration count.
Abstract Computational fluid dynamics (CFD) simulations are a vital part of the design process in the aerospace industry. Although reliable CFD results can be obtained with turbulence models, direct numerical simulation of complex bodies in three spatial dimensions (3D) is impracticable due to the massive amount of computational elements. For instance, a 3D direct numerical simulation of a turbulent boundary-layer over the wing of a commercial jetliner that resolves all relevant length scales using a serial CFD solver on a modern digital computer would take approximately 750 million years or roughly 20% of the earth’s age. Over the past 25 years, quantum computers have become the object of great interest worldwide as powerful quantum algorithms have been constructed for several important, computationally challenging problems that provide enormous speed-up over the best-known classical algorithms. In this paper, we adapt a recently introduced quantum algorithm for partial differential equations to Burgers’ equation and develop a quantum CFD solver that determines its solutions. We used our quantum CFD solver to verify the quantum Burgers’ equation algorithm to find the flow solution when a shockwave is and is not present. The quantum simulation results were compared to: (i) an exact analytical solution for a flow without a shockwave; and (ii) the results of a classical CFD solver for flows with and without a shockwave. Excellent agreement was found in both cases, and the error of the quantum CFD solver was comparable to that of the classical CFD solver.
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.
This chapter describes results of a recent investigation aiming to assess the potential of quantum computing and suitably designed algorithms for future computational fluid dynamics applications.For quantum computers becoming available in the near future, it can be expected that applications of quantum computing follow the quantum coprocessor model, where selected parts of the computational task for which efficient quantum algorithms exist are executed on the quantum hardware.For example, in computational fluid dynamics algorithm, this hybrid quantum/classical approach is discussed, and in particular it is shown how the approximate quantum Fourier transform (AQFT) can be used in the Poisson solvers of the considered method for the incompressible-flow Navier-Stokes equations.The analysis shows that despite the inevitable errors introduced by applying AQFT, the method produces meaningful results for three-dimensional example problems.A second example of a quantum algorithm for flow simulations is then described.This method based on kinetic modeling of the flow was developed to reduce the information transfer between quantum and classical hardware in the quantum coprocessor model.It is shown that this quantum algorithm can be executed fully on quantum hardware during a simulation.The conclusion summarizes further challenges for algorithm developments and future work.
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.