Semantics for a Quantum Programming Language by Operator Algebras

TL;DR

This paper introduces a new denotational semantics for a quantum programming language using operator algebras, enabling the modeling of infinite structures and unified classical-quantum computation.

Contribution

It develops a semantics based on W*-algebras that provides a categorical framework for quantum programming languages, extending prior models to infinite and classical-quantum contexts.

Findings

Categorical semantics for quantum programming using W*-algebras

Unified treatment of classical and quantum computations

Supports infinite structures in quantum programming models

Abstract

This paper presents a novel semantics for a quantum programming language by operator algebras, which are known to give a formulation for quantum theory that is alternative to the one by Hilbert spaces. We show that the opposite category of the category of W*-algebras and normal completely positive subunital maps is an elementary quantum flow chart category in the sense of Selinger. As a consequence, it gives a denotational semantics for Selinger's first-order functional quantum programming language QPL. The use of operator algebras allows us to accommodate infinite structures and to handle classical and quantum computations in a unified way.