The large range of spatial and temporal scales in high-Reynolds number turbulent flows makes its computation very expensive, demanding ever larger and more resource-intensive computational centers [1]. Beyond that, the technological development described by Moore's Law is reaching its limits, effectively slowing down the increase in computing capacity. In this regard, Quantum Computers (QC) promise a potential that their classical counterparts cannot offer. The implementation of numerical methods for fluid flows in QC is the focus of our current research. We will present a Variational Quantum Algorithm (VQA) tailor-made for implementing initial-boundary value problems on a QC. To this end, the Partial Differential Equation (PDE) to be solved is cast into an optimal control problem. This comprises a corresponding objective function and it is modular with respect to the control-to-state operator. The general approach is based on a hybrid classical-quantum architecture where the objective function is evaluated efficiently on the QC while the optimization employs classical hardware [2]. A focal point of the presentation refers to a novel treatment of boundary conditions specialized to the unique properties of the QC hardware. In particular, boundaries and PDE operators, are first decomposed in a sequence of unitary operations and then compiled into quantum gates. The feasibility and the attainable accuracy of the proposed VQA is demonstrated by solving second-order PDEs. The cases under study include steady and unsteady diffusive transport equations for a scalar property with and without convective fluxes in combination with homogeneous, as well as inhomogeneous, Dirichlet, Neumann, or mixed boundary conditions. Results obtained display a robust performance and a fair predictive accuracy, which paves the way towards the advent of Quantum CFD (QCFD). REFERENCES: [1] Jaksch, D., Givi, P., Daley, A. J., and Rung, T., Variational Quantum Algorithms for Computational Fluid Dynamics. AIAA Journal 2023. [2] Lubasch, M., Joo, J., Moinier, P., Kiffner, M. and Jaksch, D., Variational quantum algorithms for nonlinear problems. Phys. Rev. A 101, 2020.