Investigating Error Mitigation & Suppression Strategies for the HHL Quantum Linear Solver in Canonical Fluid Flow Simulations

In recent years, quantum computing is increasingly being explored as a promising avenue for accelerating computational fluid dynamics simulations. In 2009, Harrow, Hassidim, and Lloyd proposed a quantum algorithm that can solve linear systems exponentially faster than the best-known classical methods. In this study, we implement the HHL-based quantum linear solver for the canonical Hele–Shaw fluid-flow problem and benchmark its performance on IBM's quantum simulators, noise-modeling quantum emulators, and real superconducting hardwares. We first perform a parametric convergence study in which, for each problem size, we vary the shot count and track the resulting solution fidelity. This enables us to pinpoint the minimum number of shots at which the solution fidelity plateaus. Fixing that converged shot count, we then examine how fidelity varies with increasing problem size on a given IBM backend. Together, these two sweeps provide a benchmark of the HHL solver across a range of grid sizes and IBM backends. To overcome hardware noise, readout errors, and gate infidelities, we systematically apply a suite of error-mitigation strategies—including Pauli twirling, zero-noise extrapolation (ZNE), M³ measurement-error mitigation, and probabilistic error cancellation (PEC)—to the HHL solver. By combining and calibrating different error-mitigation strategies, we are currently working toward improving the solution‑state fidelity of the HHL solver on moderate‑size Hele–Shaw grids. These findings highlight both the near-term potential of quantum linear-systems algorithms for fluid-flow problems and the continuing need for robust mitigation protocols on current noisy superconducting hardwares.