Q# as a Quantum Algorithmic Language

TL;DR

This paper formalizes Q# by defining an idealized version, λ-Q#, with a rigorous mathematical foundation, safety guarantees, and semantics, facilitating its evolution and understanding as a quantum programming language.

Contribution

It introduces λ-Q# as a formal model of Q#, providing a mathematical basis and safety properties for the language.

Findings

λ-Q# enforces safety properties through its type system

Semantics based on a fully complete algebraic theory by Staton

Formalization enables further language evolution

Abstract

Q# is a standalone domain-specific programming language from Microsoft for writing and running quantum programs. Like most industrial languages, it was designed without a formal specification, which can naturally lead to ambiguity in its interpretation. We aim to provide a formal language definition for Q#, placing the language on a solid mathematical foundation and enabling further evolution of its design and type system. This paper presents $\lambda$-Q#, an idealized version of Q# that illustrates how we may view Q# as a quantum Algol (algorithmic language). We show the safety properties enforced by $\lambda$-Q#'s type system and present its equational semantics based on a fully complete algebraic theory by Staton.