A functional quantum programming language

We introduce the language QML, a functional language for quantum computations on finite types. Its design is guided by its categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive semantics of irreversible quantum computations realisable as quantum gates. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings explicit. Strict programs are free from decoherence and hence preserve superpositions and entanglement - which is essential for quantum parallelism.Comment: 15 pages. Final version, to appear in Logic in Computer Science 200

Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory

We describe categorical models of a circuit-based (quantum) functional programming language. We show that enriched categories play a crucial role. Following earlier work on QWire by Paykin et al., we consider both a simple first-order linear language for circuits, and a more powerful host language, such that the circuit language is embedded inside the host language. Our categorical semantics for the host language is standard, and involves cartesian closed categories and monads. We interpret the circuit language not in an ordinary category, but in a category that is enriched in the host category. We show that this structure is also related to linear/non-linear models. As an extended example, we recall an earlier result that the category of W*-algebras is dcpo-enriched, and we use this model to extend the circuit language with some recursive types

Semantics for a Quantum Programming Language by Operator Algebras

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.Comment: In Proceedings QPL 2014, arXiv:1412.810

A lambda calculus for quantum computation with classical control

The objective of this paper is to develop a functional programming language for quantum computers. We develop a lambda calculus for the classical control model, following the first author's work on quantum flow-charts. We define a call-by-value operational semantics, and we give a type system using affine intuitionistic linear logic. The main results of this paper are the safety properties of the language and the development of a type inference algorithm.Comment: 15 pages, submitted to TLCA'05. Note: this is basically the work done during the first author master, his thesis can be found on his webpage. Modifications: almost everything reformulated; recursion removed since the way it was stated didn't satisfy lemma 11; type inference algorithm added; example of an implementation of quantum teleportation adde

QPCF: higher order languages and quantum circuits

qPCF is a paradigmatic quantum programming language that ex- tends PCF with quantum circuits and a quantum co-processor. Quantum circuits are treated as classical data that can be duplicated and manipulated in flexible ways by means of a dependent type system. The co-processor is essentially a standard QRAM device, albeit we avoid to store permanently quantum states in between two co-processor's calls. Despite its quantum features, qPCF retains the classic programming approach of PCF. We introduce qPCF syntax, typing rules, and its operational semantics. We prove fundamental properties of the system, such as Preservation and Progress Theorems. Moreover, we provide some higher-order examples of circuit encoding

Generating reversible circuits from higher-order functional programs

Boolean reversible circuits are boolean circuits made of reversible elementary gates. Despite their constrained form, they can simulate any boolean function. The synthesis and validation of a reversible circuit simulating a given function is a difficult problem. In 1973, Bennett proposed to generate reversible circuits from traces of execution of Turing machines. In this paper, we propose a novel presentation of this approach, adapted to higher-order programs. Starting with a PCF-like language, we use a monadic representation of the trace of execution to turn a regular boolean program into a circuit-generating code. We show that a circuit traced out of a program computes the same boolean function as the original program. This technique has been successfully applied to generate large oracles with the quantum programming language Quipper.Comment: 21 pages. A shorter preprint has been accepted for publication in the Proceedings of Reversible Computation 2016. The final publication is available at http://link.springer.co

An Algebra of Pure Quantum Programming

We develop a sound and complete equational theory for the functional quantum programming language QML. The soundness and completeness of the theory are with respect to the previously-developed denotational semantics of QML. The completeness proof also gives rise to a normalisation algorithm following the normalisation by evaluation approach. The current work focuses on the pure fragment of QML omitting measurements.Comment: To appear in ENTCS, 3rd International Workshop on Quantum Programming Languages, 2005. 21 Page