  The application of quantum algorithms to classical problems is generally
accompanied by significant bottlenecks when transferring data between quantum
and classical states, often negating any intrinsic quantum advantage. Here we
address this challenge for a well-known algorithm for linear systems of
equations, originally proposed by Harrow, Hassidim and Lloyd (HHL), by adapting
it into a predictor-corrector instead of a direct solver. Rather than seeking
the solution at the next time step, the goal now becomes determining the change
between time steps. This strategy enables the intelligent omission of
computationally costly steps commonly found in many classical algorithms, while
simultaneously mitigating the notorious readout problems associated with
extracting solutions from a quantum state. Random or regularly performed skips
instead lead to simulation failure. We demonstrate that our methodology secures
a useful polynomial advantage over a conventional application of the HHL
algorithm. The practicality and versatility of the approach are illustrated
through applications in various fields such as smoothed particle hydrodynamics,
plasma simulations, and reactive flow configurations. Moreover, the proposed
algorithm is well suited to run asynchronously on future heterogeneous hardware
infrastructures and can effectively leverage the synergistic strengths of
classical as well as quantum compute resources.
