Enriched Categories for Parameterized Circuit Semantics

It is well-known that combinatorial circuits are modeled mathematically by string diagrams in a monoidal category. Given a gate set $\Sigma$, the circuits over $\Sigma$ can be thought of as string diagrams in the free monoidal category generated by $\Sigma$. In this model, circuit semantics are then given by monoidal functors out of this free category. For quantum circuits, this functor is often valued in the category of unitary matrices. This model suffices for concrete quantum circuits, but fails to describe parameterized families of quantum circuits, such as those which arise in the analysis of ansatz circuits. Intuitively, this functor should be valued in parameterized families of unitary matices, though it is not immediately clear what this mean through a categorical lens. In this paper, we show that the parameterized semantics studied in prior work can be understood through enrichment and internal constructions. We determine sufficient conditions under which this construction yields a symmetric monoidal category, and suggest how these semantics could be extended to classical circuit analysis and parameterized equivalence checking.