  A new algorithm for solving the Navier-Stokes equations (NSE) on a quantum
device is presented. For the fluid flow equations the stream function-vorticity
formulation is adopted, while the lattice Boltzmann method (LBM) is utilized
for solving the corresponding system of equations numerically for one time
step. Following the nature of the lattice Boltzmann method, the proposed
quantum algorithm consists of five major sections: initialization, collision,
propagation, boundary condition implementation, and calculation of macroscopic
quantities. The collision and boundary condition step is quantumly implemented
by applying the standard-form encoding approach, while the quantum walk
procedure is applied for the propagation step. The algorithm is implemented by
using IBM's quantum computing software development framework Qiskit, while for
the verification purposes two-dimensional (2D) cavity flow is simulated and
compared with classical code.
