A Lambda Calculus for Quantum Computation

TLDR

The classical lambda calculus serves as both a programming language and a formal algebraic system equivalent to the Turing machine, providing foundational benefits to classical computation theory. The authors propose a quantum lambda calculus to enable expression and reasoning about quantum algorithms, mirroring the advantages of classical lambda calculus. They construct a quantum lambda calculus that blends features of the quantum Turing machine and quantum circuit models, providing a computational framework and equational proof system. The calculus is closely related to linear lambda calculi used in Linear Logic and is argued to be equivalent to the quantum Turing machine.

Abstract

The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a quantum lambda calculus as an alternative model of quantum computation, which combines some of the benefits of both the quantum Turing machine and the quantum circuit models. The calculus turns out to be closely related to the linear lambda calculi used in the study of Linear Logic. We set up a computational model and an equational proof system for this calculus, and we argue that it is equivalent to the quantum Turing machine.