Integrating quantum algorithms into classical frameworks: a predictor–corrector approach using HHL

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.