This submission introduces the monoidal closed category qCPO of quantum cpos, whose objects are `quantized’ analogs of omega-complete partial orders (cpos). The category qCPO is enriched over CPO, and contains both the category CPO of cpos, and the opposite of the category FdAlg of finite-dimensional operator algebras as monoidal subcategories. The category qCPO enjoys the same properties that make CPO so useful for the semantics of higher-order programming languages that support recursion. Since every finite-dimensional operator algebra is a quantum cpo, qCPO is a natural candidate for modeling higher-order quantum programming languages that support recursion. Indeed, we use qCPO to construct a sound model for the quantum programming language Proto-Quipper-M (PQM) extended with term recursion; this same model also is a sound and computationally adequate model for LNL-FPC, a circuit-free fragment of PQM with recursive types, which can also be regarded as an extension of FPC with linear types. Previously the only known adequate model for LNL-FPC was within CPO, but CPO is not a model for PQM.