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arXiv:quant-ph/0307150 · SIAM Journal on Computing

A Lambda Calculus for Quantum Computation

André van Tonder

Published Jan 1, 2004• 196 citations
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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.

DOI
10.1137/s0097539703432165

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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.
We discuss the design of a typed lambda calculus for quantum computation. After a brief discussion of the role of higher-order functions in quantum information theory, we define the quantum lambda calculus and its operational semantics. Safety invariants, such as the no-cloning property, are enforced by a static type system that is based on intuitionistic linear logic. We also describe a type inference algorithm and a categorical semantics. 4.1 Introduction The lambda calculus, developed in the 1930s by Church and Curry, is a formalism for expressing higher-order functions. In a nutshell, a higher-order function is a function that inputs or outputs a “black box,” which is itself a (possibly higher-order) function. Higher-order functions are a computationally powerful tool. Indeed, the pure untyped lambda calculus has the same computational power as Turing machines (Turing 1937). At the same time, higher-order functions are a useful abstraction for programmers. They form the basis of functional programming languages such as LISP (McCarthy 1960), Scheme (Sussman and Steele 1975), ML (Milner 1978), and Haskell (Hudak et al. 2007). In this chapter, we discuss how to combine higher-order functions with quantum computation. We believe that this is an interesting question for a number of reasons. First, the combination of higher-order functions with quantum phenomena raises the prospect of entangled functions . Certain well-known quantum phenomena can be naturally described in terms of entangled functions, and we give some examples of this in Section 4.2.
With a view towards models of quantum computation and/or the interpretation of linear logic, we define a functional language where all functions are linear operators by construction. A small step operational semantic (and hence an interpreter/simulator) is provided for this language in the form of a term rewrite system. The linear-algebraic lambda-calculus hereby constructed is linear in a different (yet related) sense to that, say, of the linear lambda-calculus. These various notions of linearity are discussed in the context of quantum programming languages. KEYWORDS: quantum lambda-calculus, linear lambda-calculus, λλλ-calculus, quantum logics.
Particle-style token machines are a way to interpret proofs and programs, when the latter are defined according to the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple linear quantum λλλ-calculus. This, however, requires generalizing the concept of a token machine to one in which more than one particle can possibly travel around the term at the same time. This is intimately related to entanglement and allows to give a simple operational semantics to the calculus coherently with the principles of quantum computation.
We show that the lambda-q calculus can efficiently simulate quantum Turing machines by showing how the lambda-q calculus can efficiently simulate a class of quantum cellular automaton that are equivalent to quantum Turing machines. We conclude by noting that the lambda-q calculus may be strictly stronger than quantum computers because NP-complete problems such as satisfiability are efficiently solvable in the lambda-q calculus but there is a widespread doubt that they are efficiently solvable by quantum computers.
Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum lambda-calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with the principles of quantum computation.
This paper studies the linear fragment of the programing language for quantum computation with classical control described in [Selinger, P. and B. Valiron, A lambda calculus for quantum computation with classical control, in: P. Urzyczyn, editor, Proceedings of the Seventh International Conference on Typed Lambda Calculi and Applications (TLCA 2005), Lecture Notes in Computer Science 3461 (2005), pp. 354–368]. We sketch the language, and discuss equivalence of terms. We also describe a fully abstract denotational semantics based on completely positive maps.