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.
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.
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.
Partial differential equations (PDEs) present important computational challenges in several scientific disciplines. Classical numerical methods, while effective, often encounter scalability issues and high computational costs, particularly for complex problems. Quantum computing seems to address these challenges, offering a potential exponential speedup over classical methods. The presented work studies the application of quantum computing, specifically Quantum Annealing, to the computational complexities inherent in solving the Navier-Stokes equations within Computational Fluid Dynamics (CFD). Our study examines the quantum annealing methodology, conducting two different experiments and analyzing the error between the simulation and the analytical solution. The results obtained demonstrate that quantum annealing paradigm can solve this problems and the errors obtained are acceptable in comparison with other methods in the state of the art. By combining theoretical insights with practical implementation, this research clarifies the potential of quantum annealing, providing a comprehensive overview of the simulation framework, analyzing the obtained results, and suggesting future research avenues.
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.
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.
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.
Computational Fluid Dynamics (CFD) plays a central role in mechanical system design by enabling the simulation of complex flow phenomena such as turbulence, phase transitions, and fluid–structure interactions. However, high-fidelity CFD simulations—particularly those involving turbulent and multiphase flows—are often limited by the computational cost of solving large-scale, nonlinear, and high-dimensional systems of equations. This study explores the potential of quantum-enhanced computational fluid dynamics, focusing on hybrid quantum–classical solvers as a next-generation approach to overcoming these limitations. By integrating quantum algorithms with classical numerical methods, hybrid frameworks aim to accelerate key CFD tasks such as linear system solving, optimization, and high-dimensional state exploration while preserving the stability and accuracy of established solvers. The paper examines how quantum principles such as superposition and entanglement can be leveraged to improve convergence rates and design-space exploration in turbulence modeling and multiphase flow simulations. Through a conceptual and methodological analysis, this work highlights the opportunities, current constraints, and future implications of quantum-enhanced CFD for mechanical system design, offering a forward-looking perspective on the convergence of quantum computing and fluid mechanics.