Selinger gave a superoperator model of a first-order quantum programming language and proved that it is fully definable and hence fully abstract. This paper proposes an extension of the superoperator model to higher-order programs based on modules over superoperators or, equivalently, enriched presheaves over the category of superoperators. The enriched presheaf category can be easily proved to be a model of intuitionistic linear logic with cofree exponential, from which one can cave out a model of classical linear logic by a kind of bi-orthogonality construction. Although the structures of an enriched presheaf category is usually rather complex, we proved that a morphism can be expressed simply as a matrix of completely positive maps in a sufficiently large subcategory of the enriched presheaf category, which subsumes the classical model. The model inherits many desirable properties from the superoperator model such as $\omega$CPO-enrichment. Interestingly it has a sufficient structure to interpret arbitrary recursive types. We introduce \emph{Quantum FPC}, a quantum $\lambda$-calculus with recursive types, and prove that our model is a fully abstract model of Quanutm FPC.