  Topological strings on toric Calabi--Yau threefolds can be defined
non-perturbatively in terms of a free Fermi gas of N particles. Using this
approach, we propose a definition of quantum mirror curves as quantum
distributions on phase space. The quantum distribution is obtained as the
Wigner transform of the reduced density matrix of the Fermi gas. We show that
the classical mirror geometry emerges in the strongly coupled, large N limit in
which hbar ~ N. In this limit, the Fermi gas has effectively zero temperature,
and the Wigner distribution becomes sharply supported on the interior of the
classical mirror curve. The quantum fluctuations around the classical limit
turn out to be captured by an improved version of the universal scaling form of
Balazs and Zipfel.
