Developing quantum algorithms to simulate fluid dynamics has become an active area of research, as accelerating fluid simulations could have significant impact in both industry and fundamental science. While many approaches have been proposed for simulating fluid dynamics on quantum computers, it is largely unclear whether these algorithms will provide speedup over existing classical approaches. In this paper we give evidence that quantum computers cannot significantly outperform classical simulations of fluid dynamics in general. We study two models of fluids: the Korteweg-de Vries (KdV) equation, which models shallow water waves, and the incompressible Euler equations, which model ideal, inviscid fluids. We show that any quantum algorithm simulating the KdV equation or the Euler equations for time $T$ requires $Ω(T^2)$ and $e^{Ω(T)}$ copies of the initial state in the worst case, respectively. These lower bounds hold for the task of preparing the final state, and similar bounds hold for history state preparation. We prove the lower bound for the KdV equation by investigating divergence of solitons. For the Euler equations, we show that instabilities enable fast state discrimination.
Developing quantum algorithms to simulate fluid dynamics has become an active area of research, as accelerating fluid simulations could have significant impact in both industry and fundamental science. While many approaches have been proposed for simulating fluid dynamics on quantum computers, it is largely unclear whether these algorithms will provide speedup over existing classical approaches. In this paper we give evidence that quantum computers cannot significantly outperform classical simulations of fluid dynamics in general. We study two models of fluids: the Korteweg-de Vries (KdV) equation, which models shallow water waves, and the incompressible Euler equations, which model ideal, inviscid fluids. We show that any quantum algorithm simulating the KdV equation or the Euler equations for time T requires Ω(T2) and eΩ(T) copies of the initial state in the worst case, respectively. These lower bounds hold for the task of preparing the final state, and similar bounds hold for history state preparation. We prove the lower bound for the KdV equation by investigating divergence of solitons. For the Euler equations, we show that instabilities enable fast state discrimination.
Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. The existing proposals for fault-tolerant quantum algorithms for CFD have almost exclusively been based on the Carleman embedding method, used to encode nonlinearities on a quantum computer. Here, we first identify several key challenges that must be addressed in order to establish quantum advantage using this method, including Carleman convergence, time-stepping requirements, condition-number scaling, and data extraction. Building on this analysis, we develop a novel algorithm for the incompressible lattice Boltzmann equation that addresses them, and provide its detailed analysis, including all potential sources of algorithmic complexity, as well as gate count estimates. We find that for an end-to-end problem, a modest quantum advantage may be preserved for selected observables in the high-error-tolerance regime. We establish a lower bound for the Reynolds number scaling of our quantum algorithm in dimension D at Kolmogorov microscale resolution with O(Re43(1+2D)×qM), where qM is a multiplicative overhead for data extraction with qM=O(Re83) for the drag force. This upper bounds the scaling improvement over classical algorithms by O(Re83D). However, our numerical investigations suggest a lower speedup, with a scaling estimate of O(Re1.936×qM) for D=2. Our results give robust evidence that small, but nontrivial, quantum advantages can be achieved in the context of CFD, and motivate the need for additional rigorous end-to-end quantum algorithm development.
Abstract There is great interest in using quantum computers to efficiently simulate a quantum system’s dynamics as existing classical computers cannot do this. Little attention, however, has been given to quantum simulation of a classical nonlinear continuum system such as a viscous fluid even though this too is hard for classical computers. Such fluids obey the Navier–Stokes nonlinear partial differential equations, whose solution is essential to the aerospace industry, weather forecasting, plasma magneto-hydrodynamics, and astrophysics. Here we present a quantum algorithm for solving the Navier–Stokes equations. We test the algorithm by using it to find the steady-state inviscid, compressible flow through a convergent-divergent nozzle when a shockwave is (is not) present. We find excellent agreement between numerical simulation results and the exact solution, including shockwave capture when present. Finally, we compare the algorithm’s computational cost to deterministic and random classical algorithms and show that a significant speed-up is possible. Our work points to a large new application area for quantum computing with substantial economic impact, including the trillion-dollar aerospace industry, weather-forecasting, and engineered-plasma technologies.
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.
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.
Abstract The focus of this paper is to address the structured development of numerical algorithms for water wave simulations which might execute on future quantum computers. To carry out this goal we address the feasibility of numerically simulating nonlinear wave equations in the quantum domain. Many researchers express the opinion that quantum computers may be available in the next 5 to 10 years. It therefore seems appropriate to begin preparation of the key ideas and mathematical structure that would support the development of quantum algorithms and the writing of an actual code on a hypothetical quantum computer, although the latter goal will be addressed in later papers. A number of efforts in the direction of quantum computation have already been made and some standardization of the notation has occurred. In particular Google, IBM and others have made some progress in this regard to hardware and a number of investigators have made progress with algorithms. The achievement of “quantum supremacy” has recently been advertised by Google. However, it goes without saying that considerable further progress needs to be made, particularly with regard to “quantum error correction.” We do not pursue the details of quantum computers here. Instead we study the construction of the possible quantum physics for nonlinear wave equations. The goal is to extend our understanding of the quantization of nonlinear wave equations. The problem of deep-water waves begins with the nonlinear Schroedinger (NLS) equation and its physical hierarchy: This includes the NLS, the Dysthe, the Trulsen-Dysthe, the Yan Li and the Zakharov equations. A full discussion of the quantization of some of these equations and their nearby integrable counterparts is beyond the scope of the present paper and will be presented at a later date. Here we study the simplest nonlinear, integrable wave equation, that found by Korteweg and deVries [1895]. The approach discussed herein extends the method of Heisenberg from nonlinear Hamiltonian ordinary differential equations (odes) to Hamiltonian nonlinear partial differential equations (pdes) by the development of the “matrix mechanics” appropriate for a particular nonlinear wave equation. I focus at first on the integrable Hamiltonian system for KdV and then extend this to a perturbed Hamiltonian system. The beauty of the approach is that a Hamiltonian nonlinear pde can be solved with quasiperiodic Fourier series [Osborne, 2023], an idea already used by Heisenberg for nonlinear odes in his book of 1930 [Heisenberg, 1930]. Indeed, it was his use of these Fourier series that led to matrix mechanics. This paper emphasizes the mathematical and physical structure of the quantum mechanics of the “classical integrable Hamiltonian equations” of water wave motion. We address a particular water wave equation, the Korteweg-deVries (KdV) equation, which we “quantize” using the method of Heisenberg [1930]. This leads to the “matrix mechanics” of the KdV equation: We thus find the fundamental equations of quantum mechanics which form the starting point for the development of a quantum algorithm for solving the KdV equation.
Numerical simulation of turbulent fluid dynamics needs to either parametrize turbulence—which introduces large uncertainties—or explicitly resolve the smallest scales—which is prohibitively expensive. Here, we provide evidence through analytic bounds and numerical studies that a potential quantum speedup can be achieved to simulate fluid dynamics using quantum computing. Specifically, we provide a lattice Boltzmann formulation of fluid dynamics for which we give evidence that low-order Carleman linearization is much more accurate than previously believed for these systems. This is achieved via a combination of reformulating the Navier-Stokes nonlinearity ( u · ∇ u ) to lattice-Boltzmann nonlinearity ( u 2 ) and accurately linearizing the dynamical equations, which effectively trades nonlinearity for additional degrees of freedom that add negligible expense in the quantum solver. Based on this, we apply a quantum algorithm for simulating the Carleman-linearized lattice Boltzmann equation and provide evidence that its cost scales logarithmically with system size compared with polynomial scaling in the best known classical algorithms. In this paper, we suggest that a quantum advantage may exist for simulating fluid dynamics, paving the way for simulating nonlinear multiscale transport phenomena in a wide range of disciplines using quantum computing.