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.

中文翻译：

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丰富范畴论中的经典控制、量子电路和线性逻辑

我们描述了基于电路（量子）函数式编程语言的分类模型。我们表明，丰富的类别起着至关重要的作用。继 Paykin 等人在 QWire 上的早期工作之后，我们考虑了一种用于电路的简单一阶线性语言和一种更强大的宿主语言，这样电路语言就嵌入在宿主语言中。我们对宿主语言的分类语义是标准的，涉及笛卡尔封闭类别和单子。我们不是在普通类别中解释电路语言，而是在宿主类别中丰富的类别中解释电路语言。我们表明这种结构也与线性/非线性模型有关。作为一个扩展的例子，我们回想起之前的一个结果，即 W*-代数的类别是 dcpo 丰富的，我们使用这个模型来扩展具有一些递归类型的电路语言。