  Many claims of computational advantages have been made for quantum computing
over classical, but they have not been demonstrated for practical problems.
Here, we present algorithms for solving time-dependent PDEs governing fluid
flow problems. We build on an idea based on linear combination of unitaries to
simulate non-unitary, non-Hermitian quantum systems, and generate hybrid
quantum-classical algorithms that efficiently perform iterative matrix-vector
multiplication and matrix inversion operations. These algorithms lead to
low-depth quantum circuits that protect quantum advantage, with the best-case
asymptotic complexities that are near-optimal. We demonstrate the performance
of the algorithms by conducting: (a) ideal state-vector simulations using an
in-house, high performance, quantum simulator called $\textit{QFlowS}$; (b)
experiments on a real quantum device (IBM Cairo); and (c) noisy simulations
using Qiskit Aer. We also provide device specifications such as error-rates
(noise) and state sampling (measurement) to accurately perform convergent flow
simulations on noisy devices.
