  Quantum $L_\infty$ algebras are a generalization of $L_\infty$ algebras with
a scalar product and with operations corresponding to higher genus graphs. We
construct a minimal model of a given quantum $L_\infty$ algebra via the
homological perturbation lemma and show that it's given by a Feynman diagram
expansion, computing the effective action in the finite-dimensional
Batalin-Vilkovisky formalism. We also construct a homotopy between the original
and this effective quantum $L_\infty$ algebra.
