Parameterized Quantum Circuit Semantics Through Enriched Categories

2026-07-17

AI Summary

This work addresses the limitations of traditional monoidal-categorical semantics for quantum circuits, which struggle to adequately model parametrized families of quantum circuits such as variational quantum circuits. It introduces enriched category theory to establish a unified semantic framework for parameterized quantum computation. By leveraging tools including free monoidal categories, monoidal functors, and closed monoidal structures, the approach formally captures parametrized circuits and integrates two complementary perspectives on quantum control. The resulting model rigorously supports semantic analysis of both variational circuits and controlled operations, while providing a general construction applicable to both Cartesian and closed monoidal parameter settings. This framework thus delivers a cohesive and formal understanding of the mechanisms underlying parameterized quantum computation.

Abstract

It is well-known that combinatorial circuits are modeled mathematically by string diagrams in monoidal categories. Given a gate set $Σ$, the circuits over $Σ$ can be thought of as string diagrams in the free monoidal category generated by $Σ$. 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. In this paper, we introduce an approach to parameterized circuit semantics, which is based on enriched category theory. We first introduce an abstract categorical construction, and use this to gain new insights on controlled operations and quantum communication. We then study the special cases of Cartesian monoidal parameters and monoidal closed parameters, both endowing the parameterized semantics with useful constructions.We conclude by showing that the monoidal closed case can be used to unify two perspectives on quantum control.

Problem

• parameterized quantum circuits
• circuit semantics
• enriched categories
• quantum control
• monoidal categories

Innovation

• parameterized quantum circuits
• enriched category theory
• monoidal categories
• quantum semantics
• quantum control