TY - JOUR
T1 - Quantum Implementation of Numerical Methods for Convection-Diffusion Equations: Toward Computational Fluid Dynamics
AU - Liu , Bofeng
AU - Zhu , Lixing
AU - Yang , Zixuan
AU - He , Guowei
JO - Communications in Computational Physics
VL - 2
SP - 425
EP - 451
PY - 2023
DA - 2023/03
SN - 33
DO - http://doi.org/10.4208/cicp.OA-2022-0081
UR - https://global-sci.org/intro/article_detail/cicp/21494.html
KW - Quantum computing, partial differential equations, computational fluid dynamics,
finite difference, finite element.
AB - <p style="text-align: justify;">We present quantum numerical methods for the typical initial boundary
value problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVP
is discretized into a series of linear systems via finite difference methods and explicit
time marching schemes. To solve these discrete systems in quantum computers, we
design 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 vector
to take advantage of quantum algorithms in space complexity. In the following three
stages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmetic
calculations in quantum amplitudes are also realized to sum up the increments from
these stages. The proposed quantum algorithm is implemented within the open-source
quantum computing framework Qiskit [2]. By simulating one-dimensional transient
problems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokes
equations, we demonstrate the capability of quantum computers in fluid dynamics.</p>