Quantum Implementation of Numerical Methods for Convection-Diffusion Equations: Toward Computational Fluid Dynamics (Communications in Computational Physics, 2023)
We present quantum numerical methods for the typical initial boundaryvalue problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVPis discretized into a series of linear systems via finite difference methods and explicittime marching schemes. To solve these discrete systems in quantum computers, wedesign a series of quantum circuits, including four stages of encoding, amplification,adding source terms, and incorporating boundary conditions. In the encoding stage,the initial condition is encoded in the amplitudes of quantum registers as a state vectorto take advantage of quantum algorithms in space complexity. In the following threestages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmeticcalculations in quantum amplitudes are also realized to sum up the increments fromthese stages. The proposed quantum algorithm is implemented within the open-sourcequantum computing framework Qiskit [2]. By simulating one-dimensional transientproblems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokesequations, we demonstrate the capability of quantum computers in fluid dynamics.
Keywords: Quantum computing; partial differential equations; computational fluid dynamics; finite difference; finite element.