Computation of Nonlinear Wave Motion Using a Quantum Algorithm

           

The development of quantum computers over the next decade or so suggests that the geophysical sciences may benefit from very rapid computations from “quantum supremacy.” I have developed a pilot project which would help orient researchers to the use of quantum computers. The first step, and the main topic of my talk, would be to quantize a nonlinear wave equation in order that quantum algorithms might be developed. I focus on the nonlinear Schroedinger equation (NLS). The main emphasis is to show that the NLS equation for spatially periodic boundary conditions is a Hamiltonian system: Thus, I derive the solution and the coordinates and momenta in terms of quasiperiodic Fourier series. Then I apply the method of Heisenberg to develop the matrix mechanics of the NLS equation. Quantization arises as the lack of commutation for the product of the coordinate and the momenta matrices of the equation. I also discuss other equations due to the Dysthe, Trulsen and Dysthe, Yan Li and the Zakharov equations. I discuss how the method of matrix mechanics as applied to nonlinear wave equations might be programmed on a quantum computer.