Toward Quantum Algorithms for Simulating Nonlinear Ocean Surface Waves

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.