Quantum Grothendieck rings as quantum cluster algebras

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category $O$ of representations of the quantum loop algebra introduced by Hernandez-Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type $A$ , we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite-dimensional representations of the associated quantum affine algebra. In type $A_1$ , we identify remarkable relations in this quantum Grothendieck ring.