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arXiv:2607.16114

Parameterized Quantum Circuit Semantics Through Enriched Categories

Neil J. Ross, Scott Wesley

quant-phcs.LOmath.CTSubmitted Jul 17, 2026• Announced Jul 20, 2026• 0 citations
v1 · Jul 17, 2026v2 · Jul 20, 2026
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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.

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It is well-known that combinatorial circuits are modeled mathematically by string diagrams in monoidal categories. Given a gate set ΣΣΣ, the circuits over ΣΣΣ can be thought of as string diagrams in the free monoidal category generated by ΣΣΣ. 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.
It is well-known that combinatorial circuits are modeled mathematically by string diagrams in a monoidal category. Given a gate set ΣΣΣ, the circuits over ΣΣΣ can be thought of as string diagrams in the free monoidal category generated by ΣΣΣ. 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.
We describe categorical models of a circuit-based (quantum) functional programming language. We show that enriched categories play a crucial role. Following earlier work on QWire by Paykin et al., we consider both a simple first-order linear language for circuits, and a more powerful host language, such that the circuit language is embedded inside the host language. Our categorical semantics for the host language is standard, and involves cartesian closed categories and monads. We interpret the circuit language not in an ordinary category, but in a category that is enriched in the host category. As an extended example, we recall an earlier result that the category of W*-algebras is dcpo-enriched, and we use this model to extend the circuit language with some recursive types.
We describe categorical models of a circuit-based (quantum) functional programming language. We show that enriched categories play a crucial role. Following earlier work on QWire by Paykin et al., we consider both a simple first-order linear language for circuits, and a more powerful host language, such that the circuit language is embedded inside the host language. Our categorical semantics for the host language is standard, and involves cartesian closed categories and monads. We interpret the circuit language not in an ordinary category, but in a category that is enriched in the host category. We show that this structure is also related to linear/non-linear models. As an extended example, we recall an earlier result that the category of W*-algebras is dcpo-enriched, and we use this model to extend the circuit language with some recursive types.
Quipper is a practical programming language for describing families of quantum circuits. In this paper, we formalize a small, but useful fragment of Quipper called Proto-Quipper-M. Unlike its parent Quipper, this language is type-safe and has a formal denotational and operational semantics. Proto-Quipper-M is also more general than Quipper, in that it can describe families of morphisms in any symmetric monoidal category, of which quantum circuits are but one example. We design Proto-Quipper-M from the ground up, by first giving a general categorical model of parameters and state. The distinction between parameters and state is also known from hardware description languages. A parameter is a value that is known at circuit generation time, whereas a state is a value that is known at circuit execution time. After finding some interesting categorical structures in the model, we then define the programming language to fit the model. We cement the connection between the language and the model by proving type safety, soundness, and adequacy properties.
The aim of this thesis is to develop an abstract categorical setup in order to show that C -colored manifolds (i.e. compact closed manifolds with boundary where each boundary component is colored with an object of a semisimple strongly ribbon category) behaves basically in a similar manner as quantum circuits under the action of a unitary modular functor. There, the set of gates is composed only of braid operations, rotations and Dehn-twists. We first introduce the basic mathematical structure of a quantum circuit. We then provide a complete development of a 2-dimensional CW-complex over an extended surface. Furthermore, we provide a complete development of the categorical framework in order to construct a C -extended unitary modular functor (UMF) acting from the category of C -colored surfaces and morphisms of C -colored surfaces to the category of finite-dimensional vector spaces and linear isomorphisms. We then conclude by giving a complete semantics for topological quantum computation including an abstract version of the inner product, basic data units, basic data transformations, projectors and the notion of topological invariance of the algorithms.
Quipper is a practical programming language for describing families of quantum circuits. In this paper, we formalize a small, but useful fragment of Quipper called Proto-Quipper-M. Unlike its parent Quipper, this language is type-safe and has a formal denotational and operational semantics. Proto-Quipper-M is also more general than Quipper, in that it can describe families of morphisms in any symmetric monoidal category, of which quantum circuits are but one example. We design Proto-Quipper-M from the ground up, by first giving a general categorical model of parameters and state. The distinction between parameters and state is also known from hardware description languages. A parameter is a value that is known at circuit generation time, whereas a state is a value that is known at circuit execution time. After finding some interesting categorical structures in the model, we then define the programming language to fit the model. We cement the connection between the language and the model by proving type safety, soundness, and adequacy properties.