  Let $Q$ be a non-degenerated even lattice, let $V_Q$ be the lattice vertex
algebra associated to $Q$, and let $V_Q^\eta$ be a quantum lattice vertex
algebra. In this paper, we prove the equivalence between the category
$V_Q$-modules and the category of $V_Q^\eta$-modules. As a consequence, we show
that every $V_Q^\eta$-module is completely reducible, and the set of simple
$V_Q^\eta$-modules are in one-to-one correspondence with the set of cosets of
$Q$ in its dual lattice.
