  Computational fluid dynamics is both a thriving research field and a key tool
for advanced industry 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. We present a
full-stack method to solve for incompressible fluids with memory and runtime
scaling poly-logarithmically 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 arbitrary
geometries, with non-trivial boundary conditions, and self-consistent in that
it can retrieve the solution directly from the compressed encoding, i.e.
without ever passing through the expensive dense-vector representation. This
machinery lays the foundations for a new generation of potentially radically
more efficient solvers of real-life fluid problems.
