We develop a linear logical framework within the Hybrid system and use it to reason about the type system of a quantum lambda calculus. In particular, we consider a practical version of the calculus called Proto-Quipper, which contains the core of Quipper. Quipper is a quantum programming language under active development and recently has gained much popularity among the quantum computing communities. Hybrid is a system that is designed to support the use of higher-order abstract syntax for representing and reasoning about formal systems implemented in the Coq Proof Assistant. In this work, we extend the system with a linear specification logic (SL) in order to reason about the linear type system of Quipper. To this end, we formalize the semantics of Proto-Quipper by encoding the typing and evaluation rules in the SL, and prove type soundness.

中文翻译：

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线性逻辑中 Quipper 量子编程语言元理论的形式化

我们在混合系统中开发了一个线性逻辑框架，并用它来推理量子 lambda 演算的类型系统。特别地，我们考虑了称为 Proto-Quipper 的微积分的实用版本，它包含 Quipper 的核心。Quipper 是一种正在积极开发的量子编程语言，最近在量子计算社区中广受欢迎。Hybrid 是一个系统，旨在支持使用高阶抽象语法来表示和推理 Coq Proof Assistant 中实现的形式系统。在这项工作中，我们使用线性规范逻辑 (SL) 扩展系统，以推理 Quipper 的线性类型系统。为此，我们通过对 SL 中的类型和评估规则进行编码来形式化 Proto-Quipper 的语义，并证明类型的合理性。