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We develop a linear logical framework within the Hybrid system and use it toreason about the type system of a quantum lambda calculus. In particular, weconsider a practical version of the calculus called Proto-Quipper, whichcontains the core of Quipper. Quipper is a new quantum programming languageunder active development and recently has gained much popularity among thequantum computing communities. Hybrid is a system that is designed to supportthe use of higher-order abstract syntax (HOAS) for representing and reasoningabout formal systems implemented in the Coq Proof Assistant. In this work, weextend the system with a linear specification logic (SL) in order to reasonabout the linear type system of Quipper. To this end, we formalize thesemantics of Proto-Quipper by encoding the typing and evaluation rules in theSL, and prove type soundness.

Formalization of Metatheory of the Quipper Quantum Programming Language in a Linear Logic