Semantics of Higher-Order Quantum Computation via Geometry of Interaction

TL;DR

This paper develops a novel denotational semantics for a high-level quantum programming language using Geometry of Interaction, supporting full language features and proving semantic adequacy.

Contribution

It introduces the first comprehensive interaction-based semantics for a quantum functional language with features like recursion and the ! modality.

Findings

Semantic model supports full language features

Proves adequacy of the semantics

Extends classical techniques to quantum setting

Abstract

While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages by providing interaction-based semantics of a functional quantum programming language; the latter is, much like Selinger and Valiron's, based on linear lambda calculus and equipped with features like the ! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language; we prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques---largely due to their categorical formulation---is exploited for our move from classical to quantum.