  Fluid flow simulations marshal our most powerful computational resources. In
many cases, even this is not enough. Quantum computers provide an opportunity
to speedup traditional algorithms for flow simulations. We show that
lattice-based mesoscale numerical methods can be executed as efficient quantum
algorithms due to their statistical features. This approach revises a quantum
algorithm for lattice gas automata to eliminate classical computations and
measurements at every time step. For this, the algorithm approximates the qubit
relative phases and subtracts them at the end of each time step, removing a
need for repeated measurements and state preparation (encoding). Phases are
evaluated using the iterative phase estimation algorithm and subtracted using
single-qubit rotation phase gates. This method delays the need for measurement
to the end of the computation, so the algorithm is better suited for modern
quantum hardware. We also demonstrate how the checker-board deficiency that the
D1Q2 scheme presents can be resolved using the D1Q3 scheme. The algorithm is
validated by simulating two canonical PDEs: the diffusion and Burgers'
equations on different quantum simulators. We find good agreement between
quantum simulations and classical solutions of the presented algorithm.

Computational Physics;Fluid Dynamics