Computational fluid dynamics is both an active research field and a key tool
for industrial applications. The central challenge is to simulate turbulent
flows in complex geometries, a compute-power intensive task due to the large
vector dimensions required by discretized meshes. Here, we propose a full-stack
solver for incompressible fluids with memory and runtime scaling
polylogarithmically in the mesh size. Our framework is based on matrix-product
states, a powerful compressed representation of quantum states. It is complete
in that it solves for flows around immersed objects of diverse geometries, with
non-trivial boundary conditions, and can retrieve the solution directly from
the compressed encoding, i.e. without ever passing through the expensive
dense-vector representation. These developments provide a toolbox with
potential for radically more efficient simulations of real-life fluid problems