Numerical simulations of physical systems are found in many industries, as they currently play a crucial role in product development. There are many numerical methods for solving differential equations that describe the underlying physics behind the mathematical models in the simulation, among which, the finite element method (FEM) is one of the most commonly used. Although in many applications the FEM seems to provide an acceptable solution to the problem, there are still many complex real-life processes that can be challenging to simulate numerically due to their complexity and large size. Recently, there has been a shift in research towards efficiently applying quantum algorithms in finite element analysis (FEA), as the potential and speedup that they could offer have been shown, but little to no effort has been made towards the applicability and cost efficiency of these algorithms in real-world quantum devices. In this paper, we propose a cost-efficient method for applying quantum algorithms in FEA for industrial problems post-processed by classical algorithms in order to address the limitations of available quantum hardware and their cost when accessing them through different cloud-based services. We carry this out by approximating the solution of the initially large system with a suitable quantum algorithm and using the obtained solutions to generate a set of reduced-order models (ROMs) that are much smaller in complexity and size than the original model. This allows the simulation of the original model with different parameter sets and excitations to be run efficiently on classical computers without having the need to access quantum subroutines again. This way, we have reduced the usage of quantum hardware (and thus the development cost) while still taking advantage of its quantum speedup.
Numerical simulations of physical systems are found in many industries, as they currently play a crucial role in product development. There are many numerical methods for solving differential equations that describe the underlying physics behind the mathematical models in the simulation, among which, the finite element method (FEM) is one of the most commonly used. Although in many applications the FEM seems to provide an acceptable solution to the problem, there are still many complex real-life processes that can be challenging to simulate numerically due to their complexity and large size. Recently, there has been a shift in research towards efficiently applying quantum algorithms in finite element analysis (FEA), as the potential and speedup that they could offer have been shown, but little to no effort has been made towards the applicability and cost efficiency of these algorithms in real-world quantum devices. In this paper, we propose a cost-efficient method for applying quantum algorithms in FEA for industrial problems post-processed by classical algorithms in order to address the limitations of available quantum hardware and their cost when accessing them through different cloud-based services. We carry this out by approximating the solution of the initially large system with a suitable quantum algorithm and using the obtained solutions to generate a set of reduced-order models (ROMs) that are much smaller in complexity and size than the original model. This allows the simulation of the original model with different parameter sets and excitations to be run efficiently on classical computers without having the need to access quantum subroutines again. This way, we have reduced the usage of quantum hardware (and thus the development cost) while still taking advantage of its quantum speedup.
The accurate simulation of turbulent compressible fluids is a major computational challenge for the aviation industry and possibly a key technology for improving the efficiency of future aircraft so that the ambitious climate targets can be achieved. Even though great progress has been made in the field of CFD in recent decades, and simulation software is available to a wide range of users from both commercial providers and research institutes, it still has two major shortcomings: a) The accuracy of the results is usually limited by the quality of the turbulence models. Even very advanced turbulence models such as Reynolds-Stress models are not able to accurately reproduce a wide range of flows, in particular at the border of the flight envelope. For unseen test cases involving complex physical phenomena or unconventional aircraft configurations, selecting an appropriate turbulence model is often challenging and not straightforward. b) In order to overcome the shortcomings resulting from turbulence modeling, scale-resolving methods such as large eddy simulations or even direct numerical simulations are increasingly being carried out. However, the cost of these simulations for flight Reynolds numbers is so high that even under very optimistic assumptions about the development of future HPC resources and advances in algorithms, full-aircraft simulations seem unrealistic. Even though the possibilities for the use of quantum computers in CFD are still limited due to their current stage of development, they may offer the potential to accelerate calculations in such a way that scale-resolving methods for complete aircraft seem possible. Since algorithms that can be efficiently implemented on quantum computers for solving compressible nonlinear transport equations are also largely not yet available, the time is now to deal with them.
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.
Partial differential equations (PDEs) play a central role in scientific and engineering analysis, with applications spanning fluid dynamics, heat and mass transfer, electromagnetism, quantum mechanics, and financial modeling, where they are used to describe diffusion processes, wave propagation, and the evolution of complex systems over space and time. Solving multidimensional partial differential equations (PDEs) is a computationally challenging problem, even for the most advanced classical systems. Over the past decade, quantum computing has attracted significant interest as a potential approach for solving complex computational problems, including multidimensional PDEs. Although a variety of approaches have been proposed for solving PDEs, most of the existing techniques are based on variational quantum algorithms (VQAs). Despite being promising, these VQA-based approaches suffer from low accuracy, long execution times, and limited scalability. In this work, we propose a scalable and efficient quantum algorithm for solving multidimensional PDEs. Our algorithm has two variants. One variant is based on the finite difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, whereas the other is based on FDM, C2Q, and column-by-column decomposition (CCD). We have also evaluated our algorithm using several practical case studies; namely, Poisson, heat, Black–Scholes, and Navier–Stokes equations. The results show that our proposed approach achieves higher accuracy, greater scalability, and faster execution time than the VQA-based approaches. We validated these findings on both noise-free and noisy simulators, as well as on a hardware emulator and real IBM quantum hardware.
This paper explores the application of quantum-inspired algorithms to accelerate the simulation of fluid dynamics. Traditional computational fluid dynamics (CFD) methods, particularly those solving the Navier-Stokes equations, are often computationally expensive, limiting their applicability in real-time simulations and complex scenarios. This research investigates the potential of mimicking quantum phenomena, specifically wave interference and amplitude estimation, through the design and implementation of novel algorithms. The core idea is to leverage techniques inspired by quantum computing to achieve a significant speedup in solving these equations. We present a framework utilizing amplitude estimation to efficiently approximate solutions, offering a potential pathway to overcome the computational bottlenecks inherent in conventional CFD. The presented approach provides a novel methodology for simulating fluid dynamics with reduced computational cost, opening opportunities for advancements in areas such as aerospace, automotive, and weather forecasting.
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretizes the parameter space and finds an approximate solution by solving a large system of linear equations. Here we investigate the extent to which the finite element method can be accelerated using an efficient quantum algorithm for solving linear equations. We consider the representative general question of approximately computing a linear functional of the solution to a boundary value problem and compare the quantum algorithm's theoretical performance with that of a standard classical algorithm---the conjugate gradient method. Prior work claimed that the quantum algorithm could be exponentially faster but did not determine the overall classical and quantum run times required to achieve a predetermined solution accuracy. Taking this into account, we find that the quantum algorithm can achieve a polynomial speedup, the extent of which grows with the dimension of the partial differential equation. In addition, we give evidence that no improvement of the quantum algorithm can lead to a superpolynomial speedup when the dimension is fixed and the solution satisfies certain smoothness properties.
Quantum computing uses the physical principles of very small systems to develop computing platforms which can solve problems that are intractable on conventional supercomputers. There are challenges not only in building the required hardware but also in identifying the most promising application areas and developing the corresponding quantum algorithms. The availability of intermediate-scale noisy quantum computers is now propelling the developments of novel algorithms, with applications across a variety of domains, including in aeroscience. Variational quantum algorithms are particularly promising because they are comparatively noise tolerant and aim to achieve a quantum advantage with only a few hundred qubits. Furthermore, they are applicable to a wide range of optimization problems arising throughout the natural sciences and industry. To demonstrate the possibilities for the aeroscience community, we give a perspective on how variational quantum algorithms can be used in computational fluid dynamics. We discuss how classical problems are translated into quantum algorithms and their logarithmic scaling with problem size. For an explicit example, we apply this method to Burgers’s equation in one spatial dimension. We argue that a quantum advantage over classical computing methods could be achieved by the end of this decade if quantum hardware progresses as currently envisaged and emphasize the importance of joining up development of quantum algorithms with application-specific expertise to achieve a real-world impact.