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.
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.
In recent years, significant progress has been made in the development of quantum algorithms for linear ordinary differential equations as well as linear partial differential equations. There has not been similar progress in the development of quantum algorithms for nonlinear differential equations. In the present work, the focus is on nonlinear partial differential equations arising as governing equations in fluid mechanics. First, the key challenges related to nonlinear equations in the context of quantum computing are discussed. Then, as the main contribution of this work, quantum circuits are presented that represent the nonlinear convection terms in the Navier–Stokes equations. The quantum algorithms introduced use encoding in the computational basis, and employ arithmetic based on the Quantum Fourier Transform. Furthermore, a floating-point type data representation is used instead of the fixed-point representation typically employed in quantum algorithms. A complexity analysis shows that even with the limited number of qubits available on current and near-term quantum computers (<100), nonlinear product terms can be computed with good accuracy. The importance of including sub-normal numbers in the floating-point quantum arithmetic is demonstrated for a representative example problem. Further development steps required to embed the introduced algorithms into larger-scale algorithms are discussed.
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.
Structural mechanics is commonly modeled by (systems of) partial differential equations (PDEs). Except for very simple cases where analytical solutions exist, the use of numerical methods is required to find approximate solutions. However, for many problems of practical interest, the computational cost of classical numerical solvers running on classical, that is, silicon-based computer hardware, becomes prohibitive. Quantum computing, though still in its infancy, holds the promise of enabling a new generation of algorithms that can execute the most cost-demanding parts of PDE solvers up to exponentially faster than classical methods, at least theoretically. Also, increasing research and availability of quantum computing hardware spurs the hope of scientists and engineers to start using quantum computers for solving PDE problems much faster than classically possible. This work reviews the contributions that deal with the application of quantum algorithms to solve PDEs in structural mechanics. The aim is not only to discuss the theoretical possibility and extent of advantage for a given PDE, boundary conditions and input/output to the solver, but also to examine the hardware requirements of the methods proposed in literature.
Computational fluid dynamics lies at the heart of many issues in science and engineering, but solving the associated partial differential equations remains computationally demanding. With the rise of quantum computing, new approaches have emerged to address these challenges. In this work, we investigate the potential of quantum algorithms for solving the shallow water equations, which are, for example, used to model tsunami dynamics. By extending a linearization scheme previously developed in [Phys. Rev. Research 7, 013036 (2025)] for the Navier-Stokes equations, we create a mapping from the nonlinear shallow water equation to a linear system of equations, which, in principle, can be solved exponentially faster on a quantum device than on a classical computer. To validate our approach, we compare its results to an analytical solution and benchmark its dependence on key parameters. Additionally, we implement a quantum linear system solver based on quantum singular value transformation and study its performance in connection to our mapping. Our results demonstrate the potential of applying quantum algorithms to fluid dynamics problems and highlight necessary considerations for future developments.
A variational quantum algorithm for numerically solving partial differential equations (PDEs) on a quantum computer was proposed by Lubasch et al. [Phys. Rev. A 101, 010301(R) (2020)]. In this paper we generalize the method introduced by Lubasch et al. to cover a broader class of nonlinear PDEs as well as multidimensional PDEs and study the performance of the variational quantum algorithm on several example equations. Specifically, we show via numerical simulations that the algorithm can solve instances of the single-asset Black-Scholes equation with a nontrivial nonlinear volatility model, the double-asset Black-Scholes equation, the Buckmaster equation, and the deterministic Kardar-Parisi-Zhang equation. Our simulations use up to n=12 Ansatz qubits, computing PDE solutions with 2n grid points. We also perform proof-of-concept demonstrations with a trapped-ion quantum processor from IonQ [Nat. Commun. 10, 5464 (2019)], showing accurate computation of two representative expectation values needed for the calculation of a single time step of the nonlinear Black-Scholes equation. Through our classical simulations and demonstrations on quantum hardware, we identify and discuss several open challenges for using quantum variational methods to solve PDEs in a regime with a large number (much greater than 220) of grid points, but also a practical number of gates per circuit and circuit shots.
This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs). Product formulas are used to simulate Hamiltonians which can be expressed as a sum of terms which can each be simulated individually. By simulating each of these terms in sequence, the net effect approximately simulates the total Hamiltonian. We find that the error of product formulas can be improved by randomizing over the order in which the Hamiltonian terms are simulated. We prove that this approach is asymptotically better than ordinary product formulas and present numerical comparisons for small numbers of qubits. The ODE algorithm applies to the initial value problem for time-independent first order linear ODEs. We approximate the propagator of the ODE by a truncated Taylor series, and we encode the initial value problem in a large linear system. We solve this linear system with a quantum linear system algorithm (QLSA) whose output we perform a post-selective measurement on. The resulting state encodes the solution to the initial value problem. We prove that our algorithm is asymptotically optimal with respect to several system parameters. The PDE algorithms apply the finite difference method (FDM) to Poisson's equation, the wave equation, and the Klein-Gordon equation. We use high order FDM approximations of the Laplacian operator to develop linear systems for Poisson's equation in cubic volumes under periodic, Neumann, and Dirichlet boundary conditions. Using QLSAs, we output states encoding solutions to Poisson's equation. We prove that our algorithm is exponentially faster with respect to the spatial dimension than analogous classical algorithms. We also consider how high order Laplacian approximations can be used for simulating the wave and Klein-Gordon equations. We consider under what conditions it suffices to use Hamiltonian simulation for time evolution, and we propose an algorithm for these cases that uses QLSAs for state preparation and post-processing.