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.

This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are in-terpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive operational semantics of irreversible quantum computations, realisable as quantum circuits. The quantum circuit model is also given a formal categorical definition via the category FQC. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings, which may lead to the collapse of the quantum wavefunction, explicit. Strict programs are free from measurement, and hence preserve superpositions and entanglement. A denotational semantics of QML programs is presented, which maps QML terms

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We show that quantum computation can be decomposed in a pure classical (functional) part and an effectful part modeling probabilities and measurement. The effectful part can be modeled using a generalization of monads called arrows. Both the functional and effectful parts can be elegantly expressed in the Haskell programming language. 1

The search for a semantics for higher-order quantum computation leads naturallyto the study of categories of normed cones. In the first part of this paper, we develop the theory of continuous normed cones, and prove some of their basic properties, includinga Hahn-Banach style theorem. We then describe two different concrete *-autonomous categories of normed cones. The first of these categories is built from completelypositive maps as in the author's semantics of first-order quantum computation. The second category is a reformulation of Girard's quantum coherent spaces. We also pointout why ultimately, neither of these categories is a satisfactory model of higher-order quantum computation.

Abstract. This article is a brief and subjective survey of quantum programming language research. 1 Quantum Computation Quantum computing is a relatively young subject. It has its beginnings in 1982, when Paul Benioff and Richard Feynman independently pointed out that a quantum mechanical system can be used to perform computations [11, p.12]. Feynman’s interest in quantum computation was motivated by the fact that it is computationally very expensive to simulate quantum physical systems on classical computers. This is due to the fact that such simulation involves the manipulation is extremely large matrices (whose dimension is exponential in the size of the quantum system being simulated). Feynman conceived of quantum computers as a means of simulating nature much more efficiently. The evidence to this day is that quantum computers can indeed perform certain tasks more efficiently than classical computers. Perhaps the best-known example is Shor’s factoring algorithm, by which a quantum computer can find

We introduce the language QML, a functional language for quantum computations on finite types. QML introduces quantum data and control structures, and integrates reversible and irreversible quantum computation. QML is based on strict linear logic, hence weakenings, which may lead to decoherence, have to be explicit. We present an operational semantics of QML programs using quantum circuits, and a denotational semantics using superoperators.

Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps

We apply the notion of quantum predicate proposed by D’Hondt and Panangaden to analyze a purely quantum language fragment which describes the quantum part of a future quantum computer in Knill’s architecture. The denotational semantics, weakest precondition semantics, and weakest liberal precondition semantics of this language fragment are introduced. To help reasoning about quantum programs involving quantum loops, we extend proof rules for classical probabilistic programs to our purely quantum programs. 1

The field of the Quantum Programming Languages is growing fast since 1996. There are several contributions, especially on λ-Calculus and Functional Programming. One of the most significant works on Quantum λ-Calculus was developed by André van Tonder[4, 5]. His Calculus uses the concept of the intensively studied Linear Logic. In fact, it was enough to use of the syntax introduced by Philip Wadler[6] for untyped Linear Logic. van Tonder’s Calculus is not complete: Measurement is not in the Calculus. Peter Selinger argues[2] that it is not possible to add measurement to the van Tonder’s Calculus without making it typed. Following his ideas, Selinger and Benîot Valiron developed a Quantum Lambda Calculus in 2006[3]. The purpose of our work is to add measurement to the van Tonder’s Calculus, keeping it untyped. To ensure reversibility, van Tonder added a history track that keeps the necessary information to be able to follow the reductions in backward and obtain the original state. To avoid superpositions between the history track and the computational state, he follow the ideas of the Linear Logic. However, Measurement was not included in the calculus by delaying it to the end of the algorithms. His calculus includes constants that denotes qubits and elementary gate operations on qubits. The fist change that we needed to do, was to give syntax to the constants. Measurement cares about the shape of the qubit, so we had to made a syntax that ensures well-formed terms for qubits. As Measurement is a probabilistic operation, we took some tools from the Probabilistic λ-Calculus defined by Di Pierro, Hanking and Wiklicky[1] which gave us a simple way to have probabilistic inference rules. With that, it was possible to make an operational model that includes measurement. There are several works that extends the van Tonder’s Calculus, so, it is possible to make future works to our calculus as well. Another open question is the implications to fact that a Calculus with measurement is not able to be a part of an equational theory, as van Tonder did in his work.