We propose an exact, testable relation between quantum mechanics and topological strings. The physical intuition behind this construction has its origin in the Fermi gas approach to ABJ(M) theory, in particular in its maximal supersymmetric version. By studying the relation between this model and topological strings on local P1xP1, we propose a more general Fermi gas formulation for topological strings on generic, toric Calabi-Yau manifolds. This formulation is fully non-perturbative, background independent and leads to a new and testable duality between quantum mechanics and topological strings. Our proposal has also a surprising mathematical counterpart, it exploits mirror symmetry to relate two well known mathematical fields: spectral theory and enumerative geometry.