Quipper is a domain-specific programming language for the description of quantum circuits. Because it is implemented as an embedded language in Haskell, Quipper is a very practical functional language. However, for the same reason, it lacks a formal semantics and it is limited by Haskell's type system. In particular, because Haskell lacks linear types, it is easy to write Quipper programs that violate the non-cloning property of quantum states. In order to formalize relevant fragments of Quipper in a type-safe way, the Proto-Quipper family of research languages has been introduced over the last years. In this paper we first review Proto-Quipper-M, an instance of the Proto-Quipper family based on a categorical model for quantum circuits, which features a linear type system that guarantees that the non-cloning property holds at compile time. We then derive a tentative small-step operational semantics from the big-step semantics of Proto-Quipper-M and we prove that the two are equivalent. After proving subject reduction and progress results for the tentative semantics, we build upon it to obtain a truly small-step semantics in the style of an abstract machine, which we eventually prove to be equivalent to the original semantics.

中文翻译：

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Pro-Quipper-M的抽象机器语义学

Quipper是用于描述量子电路的特定领域编程语言。由于Quipper是作为Haskell中的嵌入式语言实现的，因此它是一种非常实用的功能语言。但是，出于相同的原因，它缺乏形式语义，并且受Haskell的类型系统限制。特别是，由于Haskell缺少线性类型，因此很容易编写违反量子态非克隆特性的Quipper程序。为了以一种类型安全的方式形式化Quipper的相关片段，最近几年引入了Proto-Quipper研究语言系列。在本文中，我们首先回顾Proto-Quipper-M，这是一个基于量子电路分类模型的Proto-Quipper系列实例，其特征在于线性类型系统，可确保在编译时保持非克隆特性。然后，我们从Proto-Quipper-M的大步语义中得出了一个试探性的小步操作语义，并证明了两者是等效的。在证明主题语义的主题缩减和进展结果之后，我们以其为基础，以抽象机的方式获得了真正的小步语义，最终证明了它与原始语义是等效的。