  A quantum algorithm for solving the advection equation by embedding the
discrete time-marching operator into Hamiltonian simulations is presented.
One-dimensional advection can be simulated directly since the central finite
difference operator for first-order derivatives is anti-Hermitian. Here, this
is extended to industrially relevant, multi-dimensional flows with realistic
boundary conditions and arbitrary finite difference stencils. A single copy of
the initial quantum state is required and the circuit depth grows linearly with
the required number of time steps, the sparsity of the time-marching operator
and the inverse of the allowable error. Statevector simulations of a scalar
transported in a two-dimensional channel flow and lid-driven cavity
configuration are presented as a proof of concept of the proposed approach.
